A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
- m = n - 8
- m = n = 68
- m = n = 78
- m = n = 65
The Correct Option is C
Solution and Explanation
Given that: (\(8\) males, \(5\) females)
Committee to be selected = \(11\) members
\(m\) = no. of ways the committee is formed with at least 6 males.
\((6M, 5F)\) or \((7M, 4F)\) or \((8M, 3F) \)
\(= ^8C_6 × ^5C_5+ ^8C_7 × ^5C_4+ ^8C_8 × ^5C_3\)
\(= 78 \)
\(n\) = no. of ways the committee is formed with atleast 3 female
\((8M, 3F)\) or \((7M, 4F)\) or \((6M, SF)\)
\(= ^8C_8 × ^5C_3 + ^8C_7 × ^5C_4 + ^8C_6 × ^5C_5 \)
\(= 10 + 40 + 28\)
\(= 78 \)
\(⇒ m = n = 78 \)
So, the correct option is (C): \(m = n = 78 \)
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Concepts Used:
Permutations
A permutation is an arrangement of multiple objects in a particular order taken a few or all at a time. The formula for permutation is as follows:
\(^nP_r = \frac{n!}{(n-r)!}\)
nPr = permutation
n = total number of objects
r = number of objects selected
Types of Permutation
- Permutation of n different things where repeating is not allowed
- Permutation of n different things where repeating is allowed
- Permutation of similar kinds or duplicate objects