# 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?

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

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

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

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

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

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

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

 Mar 07, 2014 maximum number of license plate numbers by: Staff --------------------------------------- Part IX 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. ---------------------------------------

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

 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} ---------------------------------------

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

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

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

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

 Mar 07, 2014 maximum number of license plate numbers by: Staff --------------------------------------- Part XVI Thanks for writing. Staff www.solving-math-problems.com