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Q:

Finding the number of ways to choose n distinct positive integers such that sum is given?

Question 1:

Suppose you are given a positive integer n and r. Find the number of ways to choose n distinct positive integers such that the sum of the integers is r.

My attempted solution (please help me check):

Pick r distinct positive integers from the set of n.

For the sake of argument, let A denote the set of such positive integers.

The number of ways of choosing r distinct positive integers from n is:

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

Suppose that this is correct. Since the sum of the chosen integers is r, we can partition r into n parts (I’m not familiar with the term) and each part needs to include at least one integer. So the number of ways to pick r distinct positive integers from n is:

$$\binom{n}{r} = \binom{r-1}{r}$$

Now suppose that this is also correct.

Then the number of ways to choose n distinct positive integers with sum r is:

$$\binom{r-1}{r} \cdot \binom{n}{n-r} = \binom{r-1}{r} \cdot \frac{n!}{(n-r)!}$$

Is the above correct?

Question 2:

Let $A_r$ be the number of ways to choose r distinct positive integers, where the number of chosen integers is minimum.

Find the recursive relation between $A_{r+1}$ and $A_r$.

My attempts to solve it:

Suppose we consider a small example:

$A_4 = \binom{4}{2} = 6$

$A_3 = \binom{3}{3} = 1$

$A_2 = \binom{2}{3} = 2$

Let’s assume that the solution to this recurrence relation is $A_{r+1}

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