How many different academic programs could she follow?

[lisa.]

In Jessica's first-year college program she must take nine courses including at least two science courses. If there are 5 science courses, 3 math courses 4 language courses, and 5 business courses from which to choose, how many different academic programs could she follow?

so i wrote out:

5C2 x 12C7, but it is including AT LEAST two science courses, and this is where I do not know what to do.

so thanks for helping
bye for now

 

[soroban]

Hello, lisa!

In Jessica's first-year college program she must take nine courses
including at least two science courses.
If there are 5 science courses, 3 math courses 4 language courses, and 5 business courses
from which to choose, how many different academic programs could she follow?

There are 5 Science courses and 12 Others.
There are: \(\displaystyle C(17,9)\,=\,24,310\) possible programs (with no restrictions).

The opposite of "at least two sciences courses" is "0 science or 1 science".

"0 science" means: 0 Science and 9 Others.
. . There are: \(\displaystyle \,C(5,0)\cdot C(12,9) \:=\:220\) ways.

"1 science" means: 1 Science and 8 Others.
. . There are: \(\displaystyle \,C(5,1)\cdot C(12,8)\:=\:2,475\) ways.

Hence, there are: \(\displaystyle \,220\,+\,2,475\:=\:2,695\) ways to not meet the science requirement.

Therefore, there are: \(\displaystyle \,24,310\,-\,2,695\:=\:\fbox{21,615}\) ways to meet the requirement.

 

[lisa.]

thank you

thanks thats the right answer, i get it now!

Lisa