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For the following question (which I pulled of the internet)

A five member committee is to be selected from among four Math teachers and five English teachers. In how many different ways can the committee be formed under the following circumstance?

<p>A)   Anyone is eligible to serve on the committee.</p>

<p>B)   The committee must consist of $3$ Math teachers and $2$ English teachers.</p>

<p>C)   The committee must contain at least three Math teachers.</p>

<p>D)   The committee must contain at least three English teachers. </p>

<p>(Answer     $126$,   $40$,   $45$,   $81$)</p>

How would I go about solving when the requirement is at least $3$ Math Teacher Any suggestions ? . I know that when it was $3$ Math and $2$ English teachers I simply took r=3 for math and r=2 for English in the Combination Formula.

MistyD
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1 Answers1

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Since you know how to do it for exactly $3$ math teachers, you also know how to do it for exactly $4$ math teachers. There are only $4$ math teachers, so those are the only two possibilities; you just have to add them up.

joriki
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  • I still dont get it when there are $3$ Math teachers the combination is 4 and when there are 4 math teachers the combination is 5 so when there are 2 and 1 English teacher(s) the combinations are 10 and 5 respectively – MistyD Aug 06 '12 at 07:01
  • Misty, there are 5 people total on the committee, so when there are 3 Math there are also 2 English and you have to account for the number of ways of choosing them, too. – Gerry Myerson Aug 06 '12 at 07:09
  • Okay so for 3 Math and 2 English it will be $4 \times 10$ and when there are 4 Math and 1 English it will be $5 \times 5$. How do I figure out for at least 3 Math. I am still a bit confused – MistyD Aug 06 '12 at 07:14
  • @MistyD: I don't know how you got $5\cdot5$; there are only $4$ math teachers in all, so there's only one way to choose $4$ of them, not $5$ ways. On your second question: What other possibilities except for $3$ and $4$ math teachers do you see for the combination to include at least $3$ math teachers if there are $4$ math teachers in all? – joriki Aug 06 '12 at 07:17
  • @joriki sorry that was a misprint when there are 4 Math and 1 English Teacher then its $1 \times 5$ – MistyD Aug 06 '12 at 07:22
  • What other possibilities except for 3 and 4 math teachers do you see for the combination to include at least 3 math teachers if there are 4 math teachers in all ? Cant think of one - 3 is the minimum and 4 is the maximum.. – MistyD Aug 06 '12 at 07:26
  • @MistyD: Well, there you go. If those are the only two possibilities, all you need to do is add them up. – joriki Aug 06 '12 at 07:28
  • Thanks now I get it so its (Combinations of 4 math . Combinations of 1 English) + (Combinations of 3 math . Combinations of 2 English) – MistyD Aug 06 '12 at 07:33