choosing a committee

Mar 2010
110
0
Ms. Ruiz has 13 boys and 12 girls in her class. She wants to select a committee to plan the class party.

a.) If she wants 3 boys and 3 girls on the committee, how many options does she have?

b.) If Tom and Howard, two of the 13 boys, are not speaking and will not work together, how many options for workable committees does Ms. Ruiz have?

I am pretty sure about part a being 13C3 * 12C3 = 62,920 workable comm.

Part b is what is throwing me. I am pretty sure the way to do it would be 62,920 - committees with Tom and Howard together, but I dont know how to figure that number out
 

Soroban

MHF Hall of Honor
May 2006
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Lexington, MA (USA)
Hello, ihavvaquestion!

Ms. Ruiz has 13 boys and 12 girls in her class.
She wants to select a committee to plan the class party.

(a) If she wants 3 boys and 3 girls on the committee, how many options does she have?

(b) If Tom and Howard, two of the 13 boys, are not speaking and will not work together,
how many options for workable committees does Ms. Ruiz have?

I am pretty sure about part (a) being: .\(\displaystyle (_{13}C_3)\cdot(_{12}C_3) \:= \:62,\!920\) options. . . Right!


Part (b) is what is throwing me.
I am pretty sure the way to do it would be:
. . 62,920 - (committees with Tom and Howard together) . . Correct!
but I don't know how to figure that number out.

Place both Tom and Howard on the committee.
Then we must choose one more boy from the other 11 boys
. . and 3 girls from the 12 available girls.

There are: .\(\displaystyle (_{11}C_1)\cdot(_{12}C_3) \;=\; 2,420\) committees with both Tom and Howard.

Got it?

 
Mar 2010
110
0
Hey Sorban...thanks

So there would be 13C3 * 12C3 - 11C1 * 12C3 possible committees then?

69920 - 2420 = 67500 workable committees