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Mathematics Question on Combinations

The number of ways of distributing 5050 identical things among 88 persons in such a way that three of them get 88 things each, two of them get 77 things each, and remaining 33 get 44 things each, is equal to

A

(50!)(8!)(8!)3(3!)2(7!)2(4!)3(2!)\frac{\left(50!\right)\left(8!\right)}{\left(8!\right)^{3}\left(3!\right)^{2}\left(7!\right)^{2} \left(4!\right)^{3}\left(2!\right)}

B

(50!)(8!)(8!)3(7!)2(4!)3\frac{\left(50!\right)\left(8!\right)}{\left(8!\right)^{3}\left(7!\right)^{2} \left(4!\right)^{3}}

C

(50!)(8!)3(7!)2(4!)3\frac{\left(50!\right)}{\left(8!\right)^{3}\left(7!\right)^{2} \left(4!\right)^{3}}

D

(8!)(3!)2(2!)\frac{\left(8!\right)}{\left(3!\right)^{2} \left(2!\right)}

Answer

(8!)(3!)2(2!)\frac{\left(8!\right)}{\left(3!\right)^{2} \left(2!\right)}

Explanation

Solution

Number of ways of dividing 88 persons in three groups, first having 33 persons, second having 22 persons and third having 33 persons = 8!3!2!3!\frac{8!}{3! 2! 3!}. Since all the 5050 things are identical, so, required number = (8!)(3!)2⋅(2!)\frac{\left(8!\right)}{\left(3!\right)^{2}\cdot\left(2!\right)}