Permutations and Combinations 3 Question 5
5. Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys $A$ and $B$, who refuse to be the members of the same team, is
(2019 Main, 9 Jan I)
(a) 350
(b) 500
(c) 200
(d) 300
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Answer:
Correct Answer: 5. (d)
Solution:
- Number of girls in the class $=5$ and number of boys in the class $=7$
Now, total ways of forming a team of 3 boys and 2 girls
$$ ={ }^{7} C _3 \cdot{ }^{5} C _2=350 $$
But, if two specific boys are in team, then number of ways $={ }^{5} C _1 \cdot{ }^{5} C _2=50$
Required ways, i.e. the ways in which two specific boys are not in the same team $=350-50=300$.
Alternate Method
Number of ways when $A$ is selected and $B$ is not
$$ ={ }^{5} C _2 \cdot{ }^{5} C _2=100 $$
Number of ways when $B$ is selected and $A$ is not
$$ ={ }^{5} C _2 \cdot{ }^{5} C _2=100 $$
Number of ways when both $A$ and $B$ are not selected
$$ ={ }^{5} C _3 \cdot{ }^{5} C _2=100 $$
$\therefore$ Required ways $=100+100+100=300$.
