JEE MAIN - Mathematics (2019 - 9th January Morning Slot - No. 20)
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 :
500
350
200
300
Explanation
From 5 girls 2 girls can be selected
= 5C2 ways
From 7 boys 3 boys can be selected
= 7C3 way
$$ \therefore $$ Total number of ways we can select 2 girls and 3 boys
= 5C2 $$ \times $$ 7C3 ways
When two boys A and B are chosen in a team then one more boy will be chosen from remaining 5 boys.
So, no of ways 3 boys can be chosen when A and B should must be chosen = 5C1 ways
$$ \therefore $$ Total number of ways a team of 2 girl and 3 boys can be made where boy A and B must be in the team = 5C1 $$ \times $$ 5C2 ways
$$ \therefore $$ Required number of ways
= Total number of ways $$-$$ when A and B are always included.
= 5C2 $$ \times $$ 7C3 $$-$$ 5C1 $$ \times $$ 5C2
= 300
= 5C2 ways
From 7 boys 3 boys can be selected
= 7C3 way
$$ \therefore $$ Total number of ways we can select 2 girls and 3 boys
= 5C2 $$ \times $$ 7C3 ways
When two boys A and B are chosen in a team then one more boy will be chosen from remaining 5 boys.
So, no of ways 3 boys can be chosen when A and B should must be chosen = 5C1 ways
$$ \therefore $$ Total number of ways a team of 2 girl and 3 boys can be made where boy A and B must be in the team = 5C1 $$ \times $$ 5C2 ways
$$ \therefore $$ Required number of ways
= Total number of ways $$-$$ when A and B are always included.
= 5C2 $$ \times $$ 7C3 $$-$$ 5C1 $$ \times $$ 5C2
= 300
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