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Permutations or Combinations?

by Shubham
(lck)











































A moped license plate number is composed of two letters, followed by four numbers.

Using this system (two letters followed by four numbers), how many license plates can be made without duplicating any numbers?

Comments for Permutations or Combinations?

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Mar 07, 2014
maximum number of license plate numbers
by: Staff


Answer

Part I


The maximum number of unique license plates which can be made by combining any two letters from the (26 character) English alphabet, followed by any four (1 digit) numbers can be calculated easily.

Since the order of the letters and numbers is important, we will be calculating the number of permutations possible (rather than the number of combinations possible).

However, the problem statement does not specify how to take duplicate letters and duplicate digits into account.

Here are two possibilities:

Sequence A (duplicates allowed) : any letter and any number can be repeated (used more than once) in any unique license plate number, and can be repeated in multiple (unique) license plate numbers

Sequence B (duplicates NOT allowed) : each letter and each number can be used only once in the entire list of unique license plate numbers


License plate combinations possible ?




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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part II


License plate combinations:  2 letters followed by 4 numbers




Sequence A (duplicates allowed) : any letter and any number can be repeated (used more than once) in any unique license plate number, and can be repeated in multiple (unique) license plate numbers

e.g.: The letter combinations AA, or BB, or CC, etc. for the first two letters of the license plate are valid. The letter combinations AC, or AD, or AE, etc. for the first two letters of the license plate are also valid.

The 1 digit number combinations 4400 or 4444, or 9159, etc. for the last four letters of the license plate are valid.

In addition, the problem statement makes it clear that every license plate number begins with two letters which are followed by four numbers. AM2358 is a valid license plate number. 3A5M82 is not a valid license plate number.





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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part III



Duplicate letters are valid for license plate combinations





Duplicate numbers are valid for license plate combinations problem







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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part IV


With these ground rules in mind:


There are 676 unique, 2 letter sequences which can be used as the first two letters of the license plate.

There are 26 letters in the alphabet. Therefore, you have 26 choices for the first letter.

After choosing the first letter, there are 26 choices for the second letter.

26 * 26 = 676 unique two letter permutations.



Permutations of duplicate letters for license plate combinations problem





The number of two letter permutations can also be calculated using the following formula:


permutations = nr

n = number of letters to choose from = 26

r = number of letters chosen = 2

order: important

repetitions: repetition of letters is allowed

permutations = 262


permutations = 26 * 26

permutations = 676



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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part V



These are:

{a,a} {a,b} {a,c} {a,d} {a,e} {a,f} {a,g} {a,h} {a,i} {a,j} {a,k} {a,l} {a,m} {a,n} {a,o} {a,p} {a,q} {a,r} {a,s} {a,t} {a,u} {a,v} {a,w} {a,x} {a,y} {a,z} {b,a} {b,b} {b,c} {b,d} {b,e} {b,f} {b,g} {b,h} {b,i} {b,j} {b,k} {b,l} {b,m} {b,n} {b,o} {b,p} {b,q} {b,r} {b,s} {b,t} {b,u} {b,v} {b,w} {b,x} {b,y} {b,z} {c,a} {c,b} {c,c} {c,d} {c,e} {c,f} {c,g} {c,h} {c,i} {c,j} {c,k} {c,l} {c,m} {c,n} {c,o} {c,p} {c,q} {c,r} {c,s} {c,t} {c,u} {c,v} {c,w} {c,x} {c,y} {c,z} {d,a} {d,b} {d,c} {d,d} {d,e} {d,f} {d,g} {d,h} {d,i} {d,j} {d,k} {d,l} {d,m} {d,n} {d,o} {d,p} {d,q} {d,r} {d,s} {d,t} {d,u} {d,v} {d,w} {d,x} {d,y} {d,z} {e,a} {e,b} {e,c} {e,d} {e,e} {e,f} {e,g} {e,h} {e,i} {e,j} {e,k} {e,l} {e,m} {e,n} {e,o} {e,p} {e,q} {e,r} {e,s} {e,t} {e,u} {e,v} {e,w} {e,x} {e,y} {e,z} {f,a} {f,b} {f,c} {f,d} {f,e} {f,f} {f,g} {f,h} {f,i} {f,j} {f,k} {f,l} {f,m} {f,n} {f,o} {f,p} {f,q} {f,r} {f,s} {f,t} {f,u} {f,v} {f,w} {f,x} {f,y} {f,z} {g,a} {g,b} {g,c} {g,d} {g,e} {g,f} {g,g} {g,h} {g,i} {g,j} {g,k} {g,l} {g,m} {g,n} {g,o} {g,p} {g,q} {g,r} {g,s} {g,t} {g,u} {g,v} {g,w} {g,x} {g,y} {g,z} {h,a} {h,b} {h,c} {h,d} {h,e} {h,f} {h,g} {h,h} {h,i} {h,j} {h,k} {h,l} {h,m} {h,n} {h,o} {h,p} {h,q} {h,r} {h,s} {h,t} {h,u} {h,v} {h,w} {h,x} {h,y} {h,z} {i,a} {i,b} {i,c} {i,d} {i,e} {i,f} {i,g} {i,h} {i,i} {i,j} {i,k} {i,l} {i,m} {i,n} {i,o} {i,p} {i,q} {i,r} {i,s} {i,t} {i,u} {i,v} {i,w} {i,x} {i,y} {i,z} {j,a} {j,b} {j,c} {j,d} {j,e} {j,f} {j,g} {j,h} {j,i} {j,j} {j,k} {j,l} {j,m} {j,n} {j,o} {j,p} {j,q} {j,r} {j,s} {j,t} {j,u} {j,v} {j,w} {j,x} {j,y} {j,z} {k,a} {k,b} {k,c} {k,d} {k,e} {k,f} {k,g} {k,h} {k,i} {k,j} {k,k} {k,l} {k,m} {k,n} {k,o} {k,p} {k,q} {k,r} {k,s} {k,t} {k,u} {k,v} {k,w} {k,x} {k,y} {k,z} {l,a} {l,b} {l,c} {l,d} {l,e} {l,f} {l,g} {l,h} {l,i} {l,j} {l,k} {l,l} {l,m} {l,n} {l,o} {l,p} {l,q} {l,r} {l,s} {l,t} {l,u} {l,v} {l,w} {l,x} {l,y} {l,z} {m,a} {m,b} {m,c} {m,d} {m,e} {m,f} {m,g} {m,h} {m,i} {m,j} {m,k} {m,l} {m,m} {m,n} {m,o} {m,p} {m,q} {m,r} {m,s} {m,t} {m,u} {m,v} {m,w} {m,x} {m,y} {m,z} {n,a} {n,b} {n,c} {n,d} {n,e} {n,f} {n,g} {n,h} {n,i} {n,j} {n,k} {n,l} {n,m} {n,n} {n,o} {n,p} {n,q} {n,r} {n,s} {n,t} {n,u} {n,v} {n,w} {n,x} {n,y} {n,z} {o,a} {o,b} {o,c} {o,d} {o,e} {o,f} {o,g} {o,h} {o,i} {o,j} {o,k} {o,l} {o,m} {o,n} {o,o} {o,p} {o,q} {o,r} {o,s} {o,t} {o,u} {o,v}



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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part VI



{o,w} {o,x} {o,y} {o,z} {p,a} {p,b} {p,c} {p,d} {p,e} {p,f} {p,g} {p,h} {p,i} {p,j} {p,k} {p,l} {p,m} {p,n} {p,o} {p,p} {p,q} {p,r} {p,s} {p,t} {p,u} {p,v} {p,w} {p,x} {p,y} {p,z} {q,a} {q,b} {q,c} {q,d} {q,e} {q,f} {q,g} {q,h} {q,i} {q,j} {q,k} {q,l} {q,m} {q,n} {q,o} {q,p} {q,q} {q,r} {q,s} {q,t} {q,u} {q,v} {q,w} {q,x} {q,y} {q,z} {r,a} {r,b} {r,c} {r,d} {r,e} {r,f} {r,g} {r,h} {r,i} {r,j} {r,k} {r,l} {r,m} {r,n} {r,o} {r,p} {r,q} {r,r} {r,s} {r,t} {r,u} {r,v} {r,w} {r,x} {r,y} {r,z} {s,a} {s,b} {s,c} {s,d} {s,e} {s,f} {s,g} {s,h} {s,i} {s,j} {s,k} {s,l} {s,m} {s,n} {s,o} {s,p} {s,q} {s,r} {s,s} {s,t} {s,u} {s,v} {s,w} {s,x} {s,y} {s,z} {t,a} {t,b} {t,c} {t,d} {t,e} {t,f} {t,g} {t,h} {t,i} {t,j} {t,k} {t,l} {t,m} {t,n} {t,o} {t,p} {t,q} {t,r} {t,s} {t,t} {t,u} {t,v} {t,w} {t,x} {t,y} {t,z} {u,a} {u,b} {u,c} {u,d} {u,e} {u,f} {u,g} {u,h} {u,i} {u,j} {u,k} {u,l} {u,m} {u,n} {u,o} {u,p} {u,q} {u,r} {u,s} {u,t} {u,u} {u,v} {u,w} {u,x} {u,y} {u,z} {v,a} {v,b} {v,c} {v,d} {v,e} {v,f} {v,g} {v,h} {v,i} {v,j} {v,k} {v,l} {v,m} {v,n} {v,o} {v,p} {v,q} {v,r} {v,s} {v,t} {v,u} {v,v} {v,w} {v,x} {v,y} {v,z} {w,a} {w,b} {w,c} {w,d} {w,e} {w,f} {w,g} {w,h} {w,i} {w,j} {w,k} {w,l} {w,m} {w,n} {w,o} {w,p} {w,q} {w,r} {w,s} {w,t} {w,u} {w,v} {w,w} {w,x} {w,y} {w,z} {x,a} {x,b} {x,c} {x,d} {x,e} {x,f} {x,g} {x,h} {x,i} {x,j} {x,k} {x,l} {x,m} {x,n} {x,o} {x,p} {x,q} {x,r} {x,s} {x,t} {x,u} {x,v} {x,w} {x,x} {x,y} {x,z} {y,a} {y,b} {y,c} {y,d} {y,e} {y,f} {y,g} {y,h} {y,i} {y,j} {y,k} {y,l} {y,m} {y,n} {y,o} {y,p} {y,q} {y,r} {y,s} {y,t} {y,u} {y,v} {y,w} {y,x} {y,y} {y,z} {z,a} {z,b} {z,c} {z,d} {z,e} {z,f} {z,g} {z,h} {z,i} {z,j} {z,k} {z,l} {z,m} {z,n} {z,o} {z,p} {z,q} {z,r} {z,s} {z,t} {z,u} {z,v} {z,w} {z,x} {z,y} {z,z}



There are 10000 unique, 4 digit number permutations which can be used on the license plate.

There are 10 single digit numbers: 0 , 1, 2, 3, 4, 5, 6, 7, 8, 9. Therefore, you have 10 choices for the first number.

After choosing the first number, there are 10 choices for the second number, 10 choices for the third number, and 10 choices for the fourth number.

10 * 10 * 10 * 10 = 10000 unique four number permutations.




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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part VII


Permutations of duplicate numbers for license plate combinations problem





Calculation of permutations of duplicate numbers for license plate combinations problem







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Mar 07, 2014
maximum number of license plate numbers
by: Staff


---------------------------------------




Part VIII


The number of unique, 4 digit permutations can also be calculated using the following formula:


permutations = nr

n = number of numbers to choose from = 10

r = number of numbers chosen = 4

order: important

repetitions: repetition of digits is allowed

permutations = 104


permutations = 10 * 10 * 10 * 10

permutations = 10,000




total permutations when letters and digits can be repeated

total unique license plate number permutations possible
= (maximum letter permutations) * (maximum number permutations)

maximum letter permutations = 676

maximum number permutations = 10,000



total unique license plate number permutations possible
= (676 letter permutations) * (10,000 number permutations) = 6,760,000


The maximum number of license plate number permutations can also be calculated directly

total unique license plate number permutations possible
= 26(letters) * 26(letters) * 10(digits) * 10(digits) * 10(digits) *
10(digits) = 6,760,000


The maximum number of unique license plate numbers when duplicate letters and duplicate numbers are allowed






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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part IX



Calculate the maximum number of unique license plate numbers when duplicate letters and duplicate numbers are allowed




Sequence B (duplicates NOT allowed) : each letter and each number can be used only once in the entire list of unique license plate numbers


There are 650 unique 2 letter combinations which can be used as the first two letters of the license plate.

There are 26 letters in the alphabet. Therefore, you have 26 choices for the first letter.

After choosing the first letter, there are 25 choices for the second letter (since the first letter cannot be repeated).

26 * 25 = 650 unique two letter combinations.


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Mar 07, 2014
maximum number of license plate numbers
by: Staff


---------------------------------------




Part X


Permutations of 2 letters without using duplicate letters for license plate combinations problem





The number of two letter permutations can also be calculated using the following formula:

permutations = n!/(n - r)!

n = number of letters to choose from = 26

r = number of letters chosen = 2

order: important

repetitions: no repetition of letters is allowed

permutations = 26!/(26 - 2)!


permutations = 26!/(24!)

permutations = 26 * 25

permutations = 650



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Mar 07, 2014
maximum number of license plate numbers
by: Staff


---------------------------------------




Part XI


These are:

{a,b} {a,c} {a,d} {a,e} {a,f} {a,g} {a,h} {a,i} {a,j} {a,k} {a,l} {a,m} {a,n} {a,o} {a,p} {a,q} {a,r} {a,s} {a,t} {a,u} {a,v} {a,w} {a,x} {a,y} {a,z} {b,a} {b,c} {b,d} {b,e} {b,f} {b,g} {b,h} {b,i} {b,j} {b,k} {b,l} {b,m} {b,n} {b,o} {b,p} {b,q} {b,r} {b,s} {b,t} {b,u} {b,v} {b,w} {b,x} {b,y} {b,z} {c,a} {c,b} {c,d} {c,e} {c,f} {c,g} {c,h} {c,i} {c,j} {c,k} {c,l} {c,m} {c,n} {c,o} {c,p} {c,q} {c,r} {c,s} {c,t} {c,u} {c,v} {c,w} {c,x} {c,y} {c,z} {d,a} {d,b} {d,c} {d,e} {d,f} {d,g} {d,h} {d,i} {d,j} {d,k} {d,l} {d,m} {d,n} {d,o} {d,p} {d,q} {d,r} {d,s} {d,t} {d,u} {d,v} {d,w} {d,x} {d,y} {d,z} {e,a} {e,b} {e,c} {e,d} {e,f} {e,g} {e,h} {e,i} {e,j} {e,k} {e,l} {e,m} {e,n} {e,o} {e,p} {e,q} {e,r} {e,s} {e,t} {e,u} {e,v} {e,w} {e,x} {e,y} {e,z} {f,a} {f,b} {f,c} {f,d} {f,e} {f,g} {f,h} {f,i} {f,j} {f,k} {f,l} {f,m} {f,n} {f,o} {f,p} {f,q} {f,r} {f,s} {f,t} {f,u} {f,v} {f,w} {f,x} {f,y} {f,z} {g,a} {g,b} {g,c} {g,d} {g,e} {g,f} {g,h} {g,i} {g,j} {g,k} {g,l} {g,m} {g,n} {g,o} {g,p} {g,q} {g,r} {g,s} {g,t} {g,u} {g,v} {g,w} {g,x} {g,y} {g,z} {h,a} {h,b} {h,c} {h,d} {h,e} {h,f} {h,g} {h,i} {h,j} {h,k} {h,l} {h,m} {h,n} {h,o} {h,p} {h,q} {h,r} {h,s} {h,t} {h,u} {h,v} {h,w} {h,x} {h,y} {h,z} {i,a} {i,b} {i,c} {i,d} {i,e} {i,f} {i,g} {i,h} {i,j} {i,k} {i,l} {i,m} {i,n} {i,o} {i,p} {i,q} {i,r} {i,s} {i,t} {i,u} {i,v} {i,w} {i,x} {i,y} {i,z} {j,a} {j,b} {j,c} {j,d} {j,e} {j,f} {j,g} {j,h} {j,i} {j,k} {j,l} {j,m} {j,n} {j,o} {j,p} {j,q} {j,r} {j,s} {j,t} {j,u} {j,v} {j,w} {j,x} {j,y} {j,z} {k,a} {k,b} {k,c} {k,d} {k,e} {k,f} {k,g} {k,h} {k,i} {k,j} {k,l} {k,m} {k,n} {k,o} {k,p} {k,q} {k,r} {k,s} {k,t} {k,u} {k,v} {k,w} {k,x} {k,y} {k,z} {l,a} {l,b} {l,c} {l,d} {l,e} {l,f} {l,g} {l,h} {l,i} {l,j} {l,k} {l,m} {l,n} {l,o} {l,p} {l,q} {l,r} {l,s} {l,t} {l,u} {l,v} {l,w} {l,x} {l,y} {l,z} {m,a} {m,b} {m,c} {m,d} {m,e} {m,f} {m,g} {m,h} {m,i} {m,j} {m,k} {m,l} {m,n} {m,o} {m,p} {m,q} {m,r} {m,s} {m,t} {m,u} {m,v} {m,w} {m,x} {m,y} {m,z} {n,a} {n,b} {n,c} {n,d} {n,e} {n,f} {n,g} {n,h} {n,i} {n,j} {n,k} {n,l} {n,m} {n,o} {n,p} {n,q} {n,r} {n,s} {n,t} {n,u} {n,v} {n,w} {n,x} {n,y} {n,z}



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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part XII



{o,a} {o,b} {o,c} {o,d} {o,e} {o,f} {o,g} {o,h} {o,i} {o,j} {o,k} {o,l} {o,m} {o,n} {o,p} {o,q} {o,r} {o,s} {o,t} {o,u} {o,v} {o,w} {o,x} {o,y} {o,z} {p,a} {p,b} {p,c} {p,d} {p,e} {p,f} {p,g} {p,h} {p,i} {p,j} {p,k} {p,l} {p,m} {p,n} {p,o} {p,q} {p,r} {p,s} {p,t} {p,u} {p,v} {p,w} {p,x} {p,y} {p,z} {q,a} {q,b} {q,c} {q,d} {q,e} {q,f} {q,g} {q,h} {q,i} {q,j} {q,k} {q,l} {q,m} {q,n} {q,o} {q,p} {q,r} {q,s} {q,t} {q,u} {q,v} {q,w} {q,x} {q,y} {q,z} {r,a} {r,b} {r,c} {r,d} {r,e} {r,f} {r,g} {r,h} {r,i} {r,j} {r,k} {r,l} {r,m} {r,n} {r,o} {r,p} {r,q} {r,s} {r,t} {r,u} {r,v} {r,w} {r,x} {r,y} {r,z} {s,a} {s,b} {s,c} {s,d} {s,e} {s,f} {s,g} {s,h} {s,i} {s,j} {s,k} {s,l} {s,m} {s,n} {s,o} {s,p} {s,q} {s,r} {s,t} {s,u} {s,v} {s,w} {s,x} {s,y} {s,z} {t,a} {t,b} {t,c} {t,d} {t,e} {t,f} {t,g} {t,h} {t,i} {t,j} {t,k} {t,l} {t,m} {t,n} {t,o} {t,p} {t,q} {t,r} {t,s} {t,u} {t,v} {t,w} {t,x} {t,y} {t,z} {u,a} {u,b} {u,c} {u,d} {u,e} {u,f} {u,g} {u,h} {u,i} {u,j} {u,k} {u,l} {u,m} {u,n} {u,o} {u,p} {u,q} {u,r} {u,s} {u,t} {u,v} {u,w} {u,x} {u,y} {u,z} {v,a} {v,b} {v,c} {v,d} {v,e} {v,f} {v,g} {v,h} {v,i} {v,j} {v,k} {v,l} {v,m} {v,n} {v,o} {v,p} {v,q} {v,r} {v,s} {v,t} {v,u} {v,w} {v,x} {v,y} {v,z} {w,a} {w,b} {w,c} {w,d} {w,e} {w,f} {w,g} {w,h} {w,i} {w,j} {w,k} {w,l} {w,m} {w,n} {w,o} {w,p} {w,q} {w,r} {w,s} {w,t} {w,u} {w,v} {w,x} {w,y} {w,z} {x,a} {x,b} {x,c} {x,d} {x,e} {x,f} {x,g} {x,h} {x,i} {x,j} {x,k} {x,l} {x,m} {x,n} {x,o} {x,p} {x,q} {x,r} {x,s} {x,t} {x,u} {x,v} {x,w} {x,y} {x,z} {y,a} {y,b} {y,c} {y,d} {y,e} {y,f} {y,g} {y,h} {y,i} {y,j} {y,k} {y,l} {y,m} {y,n} {y,o} {y,p} {y,q} {y,r} {y,s} {y,t} {y,u} {y,v} {y,w} {y,x} {y,z} {z,a} {z,b} {z,c} {z,d} {z,e} {z,f} {z,g} {z,h} {z,i} {z,j} {z,k} {z,l} {z,m} {z,n} {z,o} {z,p} {z,q} {z,r} {z,s} {z,t} {z,u} {z,v} {z,w} {z,x} {z,y}



There are 5040 unique, 4 digit number combinations which can be used on the license plate.

There are 10 single digit numbers: 0 , 1, 2, 3, 4, 5, 6, 7, 8, 9. Therefore, you have 10 choices for the first number.

Since none of the numbers can be repeated, after choosing the first number, there are 9 choices for the second number, 8 choices for the third number, and 7 choices for the fourth number.

10 * 9 * 8 * 7 = 5040 unique four digit combinations.





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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part XIII



Permutations of numbers for license plate combinations problem when no duplicates are allowed





Permutations of 4 single digit numbers without using any duplicate digits for license plate combinations






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Mar 07, 2014
maximum number of license plate numbers
by: Staff


---------------------------------------




Part XIV


The number of unique, 4 digit permutations can also be calculated using the following formula:

permutations = n!/(n - r)!

n = number of digits to choose from = 10

r = number of digits chosen = 4

order: important

repetitions: no repetition of single-digit numbers is allowed

permutations = 10!/(10 - 4)!


permutations = 10!/(6!)

permutations = 10 * 9 * 8 * 7

permutations = 5040



total permutations when neither letters and digits can be repeated

total unique license plate number permutations possible
= (maximum letter permutations) * (maximum number permutations)

maximum letter permutations = 650

maximum number permutations = 5,040



total unique license plate number permutations possible
= (650 letter permutations) * (5,040 number permutations) = 3,276,000


The maximum number of license plate number permutations can also be calculated directly

total unique license plate number permutations possible
= 26(letters) * 25(letters) * 10(digits) * 9(digits) * 8(digits) *
7(digits) = 3,276,000



The maximum number of unique license plate numbers when duplicate letters and duplicate numbers are not allowed







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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part XV



Calculate the maximum number of unique license plate numbers when duplicate letters and duplicate digits are not allowed





Final Answer:

maximum number of unique license plates which can be made by combining any two letters from the (26 character) English alphabet, followed by any four (1 digit) numbers

Unique License Plate numbers when letters and digits can be repeated = 6,760,000


Unique License Plate numbers when neither letters or digits can be repeated = 3,276,000




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Mar 07, 2014
maximum number of license plate numbers
by: Staff


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Part XVI



Maximum number of unique license plate number combinations made by combining two letters followed by four digits







Thanks for writing.

Staff
www.solving-math-problems.com



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