A computer is programmed to produce a random sequence of n d

A computer is programmed to produce a random sequence of n digits.

a) How many possible sequences are there?

b) How many of these sequences have each of the 10 digits appearing?

Solution

a)

Each digit can be from 0,1,2,..,9 so 10 cases

So, total 10^n sequences

b)

Case 1: n<10

No such sequences are possible

Case 2:n>=10

Choose 10 places in C(n,10) ways

And then permute the 10 digits in these 10 places in 10! ways

Remaining n-10 digits have no restrictions

SO, 10^{n-10} sequences in those digits

SO, total

10!C(n,10)10^{n-10}