Question:

A student is to answer $10$ out of $13$ questions in an examination such that the he must choose at least $4$ from the first five questions. The number of choices available to him is :

Updated On: Mar 31, 2024
  • $140$
  • $196$
  • $280$
  • $348$
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The Correct Option is B

Solution and Explanation

The number of choices available to him is $= \,^{5}C_{4}\times\,^{8}C_{6}+\,^{5}C_{5} \times\,^{8}C_{5} $ $\left[\because \,^{8}C_{6}=\,^{8}C_{2}, \,^5C_4=\,^{5}C_{1}, \,^{8}C_{5}=\,^{8}C_{3}\right]$ $=5\times\frac{8\times7}{1\times2}+1\times\frac{8\times7\times6}{1\times2\times3}$ $= 140 + 56 = 196$
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Concepts Used:

Permutations and Combinations

Permutation:

Permutation is the method or the act of arranging members of a set into an order or a sequence. 

  • In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point. 
  • A permutation is used in many events of daily life. It is used for a list of data where the data order matters.

Combination:

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

  • Combination refers to 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.