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A group of 9 people consists of 2 boys, 3 girls and 4 adults. In how many ways can a team of 4 be chosen if 2 girls are in the team?

My first answer: $^3C_2 × ^7C_2 = 63$ ways

$3\choose2$ Since there must be two girls in the team and $7\choose2$ as there are $7$ people left, $1$ girl and all adults and boys, to be chosen for the two empty places the two girls aren't occupying.

Then I thought about checking from a table

3Girls 1 adult ${3\choose3}\cdot{4\choose1}$

3Girls 1 boy ${3\choose3}\cdot{2\choose1}$

2Girls 2 boys ${3\choose2}\cdot{2\choose2}$

2Girls 2 adults ${3\choose2}\cdot{4\choose2}$

2Girls 1 adult 1boy ${3\choose2}\cdot{4\choose1}\cdot{2\choose1}$

Second answer by summing all the possible ways the answer is 51 ways

Anindya Prithvi
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Manar
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2 Answers2

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There are $6$ teams that include all $3$ girls. Your first approach counts each of them $3$ times. If the girls are $g_1,g_2$, and $g_3$, and $x$ is one of the other $6$ people, you count the team $\{g_1,g_2,g_3,x\}$ once with $\{g_1,g_2\}$ as the pair counted by your $\binom32$ factor and $g_3$ as the girl chosen with $x$, once with $\{g_1,g_3\}$ as that pair, and once with $\{g_2,g_3\}$ as that pair. Thus, you’ve counted those $6$ teams as $18$ instead of as $6$, so you’ve overcounted by $12$.

Brian M. Scott
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The reason I wrote 7c2 without any hesitation is because I know that if two girls are taken the third girl will be the one left and like that no girl would be repeated.

And this is true, no girl will be repeated but the teams having them will. They will be changing places when it's not necessary as we're working with combinations.

Here are the selection containg 3 girls lead out to prove it and help these like me who couldn't imagine it:

Teams with g1 and g2 as a must

g1, g2, g3, b1*

g1, g2, g3, b2

g1, g2, g3, a1

g1, g2, g3, a2

g1, g2, g3, a3

g1, g2, g3, a4

......(rest are the teams with only 2 girls)

Teams with g1 and g3 as a must

g1, g3, g2, b1.*

(g1, g3, g2, b1 is same as g1, g2, g3, b1 from above and so are the rest)**

g1, g3 g2, b2

g1, g3 g2, a1

g1, g3 g2, a2

g1, g3 g2, a3

g1, g3 g2, a4

......(rest are the teams with only 2 girls)

Teams with g2 and g3 as a must

g2, g3, g1, b1

g2, g3, g1, b2

g2, g3, g1, a1

g2, g3, g1, a2

g2, g3, g1, a3

g2, g3, g1, a4

......(rest are the teams with only 2 girls)

Manar
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