Problem Solving

What is the total number of permutations of n different things taken not more than p times, when each thing may be repeated any number of times?

A. n^p +1
B. n(n^p-1)
C.n^p (n+1)/(n-1)
D. n(n^p - 1)/ (n-1)
E. n^p (n+1)

is the answer B?

n different things- can be repeated in n different ways but only p-1 times
therefore n(n^p-1)

The best way to solve such questions is to substitute values for n and p.

The answer will be n+n^2+n^3+....+n^p = n(n^p-1)/n-1

gabriel wrote:The best way to solve such questions is to substitute values for n and p.

Yes, this is definitely true.

Anyway, let me try to explain the concept here.

I guess every aspirant is aware of the following basic concepts.

Total number of permutations of r things from n distinct things when repetition is not allowed is nPr

Total number of permutations of r things from n distinct things when repetition is allowed is n^r

Now the given question is : we have arrange at least 1 thing and at most p things from n things and it is clearly mentioned that repetition is allowed is allowed.

1 thing can be arranged from n things in n^1 ways
2 things can be arranged from n things in n^2 ways.
............................................
.........................................
p things can be arranged from n things in n^p ways.

So the answer is n+n^2+n^3+......+n^p

This is a G.P with common ratio n (n>1)

So the sum to p terms of this G.P is n(n^p-1)/n-1

Hence D

n items taken 1 time = n

n items taken 2 times where items can repeat is possible = n.n = n^2

so total permutation = n + n^2 + n^3 + ....

n(n^p-1)/(n-1)