SATHEE: Permutations and Combinations 2 Question 14

Permutations and Combinations 2 Question 14

14. In a high school, a committee has to be formed from a group of 6 boys $M_{1}, M_{2}, M_{3}, M_{4}, M_{5}, M_{6}$ and 5 girls $G_{1}$ , $G_{2}, G_{3}, G_{4}, G_{5}$ .

(i) Let $α_{1}$ be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.

(ii) Let $α_{2}$ be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.

(iii) Let $α_{3}$ be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.

(iv) Let $α_{4}$ be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls such that both $M_{1}$ and $G_{1}$ are NOT in the committee together.

(2018 Adv.)

	List-I	List-II
P.	The value of $α_{1}$ is	1.	136
Q.	The value of $α_{2}$ is	2.	189
R.	The value of $α_{3}$ is	3.	192
S.	The value of $α_{4}$ is	4.	200
		5.	381
		6.	461

The correct option is

(a) $P \to 4; Q \to 6; R \to 2; S \to 1$

(b) $P \to 1; Q \to 4; R \to 2; S \to 3$

(d) $P \to 4; Q \to 2; R \to 3; S \to 1$

Integer Answer Type Question

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Answer:

Correct Answer: 14. (c)

Solution:

Given 6 boys $M_{1}, M_{2}, M_{3}, M_{4}, M_{5}, M_{6}$ and 5 girls $G_{1}, G_{2}, G_{3}, G_{4}, G_{5}$

(i) $α_{1} \to$ Total number of ways of selecting 3 boys and 2 girls from 6 boys and 5 girls.

i..e, $^{6} C_{3} \times^{5} C_{2} = 20 \times 10 = 200$

$∴ α_{1} = 200$

(ii) $α_{2} \to$ Total number of ways selecting at least 2 member and having equal number of boys and girls i.e., $^{6} C_{1}^{5} C_{1} +^{6} C_{2}^{5} C_{2} +^{6} C_{3}^{5} C_{3} +^{6} C_{4}^{5} C_{4} +^{6} C_{5}^{5} C_{5}$ $= 30 + 150 + 200 + 75 + 6 = 461$

$\Rightarrow α_{2} = 461$

(iii) $α_{3} \to$ Total number of ways of selecting 5 members in which at least 2 of them girls

$i.e., \begin{aligned} ^{5} C_{2}^{6} C_{3} & +^{5} C_{3}^{6} C_{2} +^{5} C_{4}^{6} C_{1} +^{5} C_{5}^{6} C_{0} \\ = 200 + 150 + 30 + 1 = 381 \\ α_{3} & = 381 \end{aligned}$

(iv) $α_{4} \to$ Total number of ways of selecting 4 members in which at least two girls such that $M_{1}$ and $G_{1}$ are not included together.

$G_{1}$ is included $\to^{4} C_{1} \cdot^{5} C_{2} +^{4} C_{2} \cdot^{5} C_{1} +^{4} C_{3}$

$= 40 + 30 + 4 = 74$

$M_{1}$ is included $\to^{4} C_{2} \cdot^{5} C_{1} +^{4} C_{3} = 30 + 4 = 34$

$G_{1}$ and $M_{1}$ both are not included

$^{4} C_{4} +^{4} C_{3} \cdot^{5} C_{1} +^{4} C_{2} \cdot^{5} C_{2}$

$1 + 20 + 60 = 81$

$∴$ Total number $= 74 + 34 + 81 = 189$

$α_{4} = 189$

Now, $P \to 4; Q \to 6; R \to 5; S \to 2$

Hence, option (c) is correct.

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Permutations and Combinations 2 Question 14

14. In a high school, a committee has to be formed from a group of 6 boys $M_{1}, M_{2}, M_{3}, M_{4}, M_{5}, M_{6}$ and 5 girls $G_{1}$ , $G_{2}, G_{3}, G_{4}, G_{5}$ .

Integer Answer Type Question

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

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एनसीईआरटी व्यायाम वीडियो समाधान

Permutations and Combinations 2 Question 14

14. In a high school, a committee has to be formed from a group of 6 boys M1,M2,M3,M4,M5,M6 and 5 girls G1, G2,G3,G4,G5.

Integer Answer Type Question

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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14. In a high school, a committee has to be formed from a group of 6 boys $M_{1}, M_{2}, M_{3}, M_{4}, M_{5}, M_{6}$ and 5 girls $G_{1}$ , $G_{2}, G_{3}, G_{4}, G_{5}$ .