I had a math test recently but still can't find the answer to one of the questions which looks rather easy. Here it is:
You are planning a 3-day camping trip, and want to buy easy-to-cook meals from your local shop. The shop stocks 5 varieties of main course and 5 varieties of desserts. You decide to buy 3 different main courses and 3 different desserts. How many options are available to you? (You are not yet thinking about which dessert to combine with which main course.)
[A]100 [B]125 [C]243 [D]400
My logic was that for both the main course and the dessert we have ${5\choose 3} = 10$ And then for the first day we have 10*10=100, 9*9=81 for the second and 8*8=64 for the third, but that is equal to 245. I am really confused because it looks so simple, yet the answers don't match.
$\binom 53=10$ ways to choose the three main courses and $\binom 53=10$ ways to choose the three desserts.
Thus there are $10\times 10=100$ ways to select your foods.