How can we prove the formula n+r-1Cr when r things are to be selected out of n type of things - Maths - Permutations and Combinations

Question

How can we prove the formula n+r-1Cr when r things are to be
selected out of n type of things where each object may be repeated
any no of times? OR how can we justify this formula logically?

Gursheen kaur. · Accepted Answer

Here is the link similar to your query https://www meritnation com/ask-answer/question/prove-npr-n-n-r-and-deduce-it-t/binomial-theorem/3191087 Total no of things=n no of items chosen from total no of things =r âˆ´There are n ways of replacing the first object and for second , there are (n-1) ways and for rth ,there are (n-r+1) ways Thus, the total number for the order to be relevant is $P (n, r) = n (n - 1) (n - 2) (n - 3) (n - (r - 1)) \Rightarrow P (n, r) = \frac{n (n - 1) (n - 2) (n - 3) (n - (r - 1)) (n - r) (n - (r - 1)) 3 2 1}{(n - r) (n - (r - 1)) 3 2 1} \Rightarrow P (n, r) = \frac{n!}{(n - r)!}$ Now, No of ways to rearrange the group = r! ways ${C_{r}}^{n} = \frac{n!}{(n - r)! r!} \Rightarrow {C_{r}}^{n} = \frac{{P_{r}}^{n}}{r!}$

How can we prove the formula n+r-1Cr when r things are to be selected out of n type of things where each object may be repeated any no. of times? OR how can we justify this formula logically?