Question

Two n-digit sequences of digits 0,1,. . . ,9 are said to be of the same type if the digits of one are a permutation of the digits of the other. For n = 8, for example, the sequences 03088929 and 00238899 are the same type. How many types of n-digit sequences are there?

Answer 1

(n+9)!/(n!9!)

Explanation:

A combination is a selection of items from a collection, such that (unlike permutations) the order of selection does not matter.

This is a case of combination with repetition because any n-digit sequence can have numbers repeated. For example in the sequence 00238899 there are 2 zeros.

The formula to find the number of combinations when repetition is allowed is:

(n+N-1)!/[n!(N-1)!]

where

N is the number of elements of the set we can choose from

n is the number of elements we choose

In this case, the set {0,1,2,3,4,5,6,7,8,9} has 10 elements, N=10

The number of n-digit sequences is given by

(n+10-1)!/[n!(10-1)!]=(n+9)!/(n!9!)