Question

What are the number of ways in which $\left( p+q \right)$ dissimilar objects can be portioned into two groups consisting of $p$ and $q$ objects respectively, where $p\ne q$?

Answer 1

Hint: For solving this question you should know about permutation and combination. In this problem we will use these for finding the number of ways. We will use the combination in this to find our required number of ways. And we solve this to find the original digit.

Complete step by step answer:
According to our question we have to find the number of ways in which dissimilar objects can be portioned into two groups consisting of $p$ and $q$ objects respectively, where $p\ne q$. As we know, permutation and combinations are generally used for solving these types of questions. The permutation is used for lists and in the permutation the order matters and in the combination the combination is used for groups and in this the order does not matter. Permutations relate to the act of arranging all the members of a set into some sequence or order. And the combination is a way of selecting items from a collection such that (unlike permutations) the order of selection does not matter.
According to the given data in the question we can say that if we put any $p$ in 1 group automatically we will have another group with $q$ objects.
Hence the number of ways is,
${}^{p+q}{{C}_{p}}={}^{p+q}{{C}_{q}}$
So, this is the final answer.

Note: While solving these types of questions you should be careful about the selection of permutation or combination. You have to be sure which will be used in the given problem because if the wrong method is selected then the answer will also be wrong.