SATHEE: Permutations And Combinations Question 331

Permutations And Combinations Question 331

Question: The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

Options:

A) $ ^{8}C_3 $

B) 21

C) $ 3^{8} $

D) 5

Show Answer

Answer:

Correct Answer: B

Solution:

[b] Required number of ways = coefficient of $ x^{8} $ in $ {{(x+x^{2}+..x^{6})}^{3}} $ [ $ \because $ Each box can receive minimum 1 and maximum 6 balls] = coeff of $ x^{8} $ in $ x^{2}{{(1+x+x^{2}+…+x^{5})}^{3}} $ = coeff of $ x^{5} $ in $ {{( \frac{1-x^{6}}{1-x} )}^{3}} $ = coeff of $ x^{5} $ in $ {{(1-x)}^{-3}} $ = coeff of $ x^{5} $ in $ (1{{+}^{3}}C_1x{{+}^{4}}C_2x^{2}+…) $ $ {{=}^{7}}C_5=21 $

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Permutations And Combinations Question 331

Question: The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

Options:

Answer:

Solution:

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