Question

Four boys picked up 30 mangoes. In how many ways can they divide them if all mangoes are identical?

Answer 1

Hint: As we know combination determines the number of possible arrangements in a collection of items in which the order of the selection does not matter and hence, to solve the given question, apply the formula to get the number of ways if mangoes are identical and the formula is given as \[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\].

Formula used:
\[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\]
In which,
n is the total number of objects.
\[{}^n{C_r}\] is a number of combinations.
r is the number of choosing objects from the set.

Complete step-by-step solution:
 Let us write the given data:
Here, 30 mangoes are to be distributed among four boys, where each can get any number of mangoes.
Therefore, total number of ways = \[{}^{30 + 4 - 1}{C_{4 - 1}}\]
 \[\Rightarrow {}^{33}{C_3}\]
\[\Rightarrow \dfrac{{33 \times 32 \times 31}}{{1 \times 2 \times 3}}\]
\[\Rightarrow 5456\]

Hence the correct answer is 5456.

Additional information: Permutation and combination are the ways to represent a group of objects by selecting them in a set and forming subsets. It defines the various ways to arrange a certain group of data. When we select the data or objects from a certain group, it is said to be permutations, whereas the order in which they are represented is called combination.

Note: The combination formula can be used only when the objects from a set are selected without repetition. Combinations can be confused with permutations. However, in permutations, the order of the selected items is essential and we know that factorial is a product of all positive integers less or equal to the number. A permutation is used for the list of data (where the order of the data matters) and the combination is used for a group of data (where the order of data doesn’t matter).