Question:

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

Updated On: Jul 7, 2022
  • $^8C_3$
  • 21
  • $3^8$
  • 5
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The Correct Option is B

Solution and Explanation

We know that the number of ways of distributing n identical items among r persons, when each one of them receives at least one item is ${^{n-1}C_{r-1}}$ $\therefore $ The required number of ways $= ^{8-1}C_{3-1}= ^{7}C_{2} = \frac{7!}{2!5!}= \frac{7 \times6}{2 \times1} = 21 $
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Notes on Combinations

Concepts Used:

Combinations

The method of forming subsets by selecting data from a larger set in a way that the selection order does not matter is called the combination.

  • It means the combination of about ‘n’ things taken ‘k’ at a time without any repetition.
  • The combination is used for a group of data where the order of data does not matter.
  • For example, Imagine you go to a restaurant and order some soup.
  • Five toppings can complement the soup, namely:
    • croutons,
    • orange zest,
    • grated cheese,
    • chopped herbs,
    • fried noodles.

But you are only allowed to pick three.

  • There can be several ways in which you can enhance your soup with savory.
  • The selection of three toppings (subset) from the five toppings (larger set) is called a combination.

Use of Combinations:

It is used for a group of data (where the order of data doesn’t matter).