A committee of 5 members is to be formed out | vedclass

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Question

(C) To form the committee,we need to select $2$ trainees from $3$ available trainees and $3$ research associates from $6$ available research associate

Vedclass · Accepted Answer

$60$

Vedclass · Answer

(C) To form the committee,we need to select $2$ trainees from $3$ available trainees and $3$ research associates from $6$ available research associates.The number of ways to select $2$ trainees out of $3$ is given by the combination formula ${ }^{n} C_{r} = \frac{n!}{r!(n-r)!}$.${ }^{3} C_{2} = \frac{3!}{2!(3-2)!} = \frac{3 	imes 2!}{2! 	imes 1!} = 3$.The number of ways to select $3$ research associates out of $6$ is:${ }^{6} C_{3} = \frac{6!}{3!(6-3)!} = \frac{6 	imes 5 	imes 4}{3 	imes 2 	imes 1} = 20$.Since these selections are independent,the total number of ways to form the committee is the product of these two values:Total ways $= { }^{3} C_{2} 	imes { }^{6} C_{3} = 3 	imes 20 = 60$.

$A$ committee of $5$ members is to be formed out of $3$ trainees,$4$ professors,and $6$ research associates. In how many different ways can this be done if the committee must have $2$ trainees and $3$ research associates?

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