A probability question with sequences

Question

A sequence of numbers X₁, X₂,... are produced by taking a random number from the set {1,2,3...n}.

The summation S_k is defined to be X₁+X₂+...+X_k.

Which of the following is a correct expression for the probability that S_k can be smaller or equal to n?

The answer is (nCk)/nⁿ

Why? Explain

Thank you in advance! My teacher gave this to me for math extension 1 & I’m so confused, I don’t even know where to begin with this question!

Answer 1

the LARGEST the sum can be is if n is randomly selected

everytime

The sum is then n^2

THe probablitity that this happens is (1/n)^n = 1/(n^n)

There are (n choose k) ways to arrange the terms.

So the total probablity is (n choose k)/(n^n)