Question

A student must choose a program of four courses from a menu of courses consisting of English, algebra, geometry, history, art, and Latin. The ordering of the courses does not matter. How many different programs can be made if each program must have English and and least one math class.

Answer 1

there are five courses left, and you have to choose 3 of them
you are being asked to compute the number of ways you can choose 3 items out of a set of 5, called "five choose three" and written as \(\binom{5}{3}\) or sometimes \(_5C_3\) or even \(^5C_3\)
do you know how to compute this number?

Answer 2

no i do not

Answer 3

\[\binom{5}{3}=\binom{5}{2}=\frac{5\times 4}{2}\]

Answer 4

how about the last condition, English and one of the math course?

Answer 5

cancel first, multiply last

Answer 6

oops

Answer 7

I thought it was something like 1x1x6x6

Answer 8

with the two ones being the conditions, and the 6s being the available classes. Does this work?

Answer 9

one choice for english, two choices for math, and then \(4\times 3\) for the next two

Answer 10

so
\[1\times 2\times 4\times 3\]

Answer 11

oh that makes sense now. Thank you very much

Answer 12

???!!!

Answer 13

one choice for English x 2 choices for math x \(\left(\begin{matrix}3\\2\end{matrix}\right)\)
wonder where does 4 come from?