In Lisa's first-year college program she must take nine courses including at least two science courses. If there are five science courses, three mathematics courses, four language courses and five business courses from which to choose, how many different academic programs could she follow?

Without any restrictions she could take

C(17,9) different combinations or 24310

At least 2 science courses means we have to excluse
1. no science course
2. 1 science course

1. number of ways to take no science = C(5,0)xC(12,9)
= 220
2. number of ways to include 1 science course
= C(5,1)xC(12,8)
= 2475

number of ways with at least 2 science courses
= 24310 - 220 - 2475
= 21615

or

2 science courses: C(5,2)xC(12,7) = 7920
3 science courses : C(5,3)xC(12,6) = 9240
4 science courses : C(5,4)xC(12,5) = 3960
5 science courses: C(5,5)xC(12,4) = 495
Total = 21615

Thank you:) I got that answer too:)

There are 20 people in a club. In how many ways can the club elect a president, a secretary and a treasurer, on person can hold more than one office?

Well, Lisa certainly has many options to choose from! Let's calculate the number of different academic programs she could follow:

First, let's consider the number of ways Lisa can choose at least two science courses from the five available. This can be calculated using the combination formula. Therefore, Lisa can choose 2, 3, 4, or 5 science courses. Let's consider each case separately:

Case 1: Lisa chooses 2 science courses.
She can choose 2 science courses out of 5 in "5 choose 2" ways, which is calculated as: C(5,2) = 5! / (2! * (5-2)!) = 10.

Case 2: Lisa chooses 3 science courses.
She can choose 3 science courses out of 5 in "5 choose 3" ways, which is calculated as: C(5,3) = 5! / (3! * (5-3)!) = 10.

Case 3: Lisa chooses 4 science courses.
She can choose 4 science courses out of 5 in "5 choose 4" ways, which is calculated as: C(5,4) = 5! / (4! * (5-4)!) = 5.

Case 4: Lisa chooses all 5 science courses.
She can choose all 5 science courses out of 5 in "5 choose 5" ways, which is calculated as: C(5,5) = 5! / (5! * (5-5)!) = 1.

Now, for each case, let's calculate the total number of academic programs Lisa can follow:

Case 1: Lisa chooses 2 science courses.
Lisa can choose the remaining 7 courses from the remaining courses available in (3+4+5) choose 7 ways, which is calculated as: C(3+4+5,7) = (12 choose 7) = 924.

Case 2: Lisa chooses 3 science courses.
Lisa can choose the remaining 6 courses from the remaining courses available in (3+4+5) choose 6 ways, which is calculated as: C(3+4+5,6) = (12 choose 6) = 924.

Case 3: Lisa chooses 4 science courses.
Lisa can choose the remaining 5 courses from the remaining courses available in (3+4+5) choose 5 ways, which is calculated as: C(3+4+5,5) = (12 choose 5) = 792.

Case 4: Lisa chooses all 5 science courses.
Lisa doesn't have any more courses to choose from.

Now, let's calculate the total number of different academic programs Lisa could follow by summing up the possibilities for each case:

Total = Case 1 + Case 2 + Case 3 + Case 4
= 10 + 10 + 5 + 1
= 26.

So, Lisa could follow 26 different academic programs in total.

To find out how many different academic programs Lisa could follow, we need to consider the different combinations of courses she can choose.

First, let's calculate the number of combinations for the science courses. Since Lisa must take at least two science courses, we can choose any combination of 2, 3, 4, or 5 science courses. We can calculate this using the combination formula (nCr), where n is the total number of science courses (5) and r is the number of science courses to choose (2, 3, 4, or 5).

For 2 science courses:
C(5, 2) = 5! / (2!(5-2)!) = 10 combinations

For 3 science courses:
C(5, 3) = 5! / (3!(5-3)!) = 10 combinations

For 4 science courses:
C(5, 4) = 5! / (4!(5-4)!) = 5 combinations

For 5 science courses:
C(5, 5) = 5! / (5!(5-5)!) = 1 combination

Now, let's calculate the number of combinations for the remaining courses. We can choose any number of mathematics courses (0, 1, 2, or 3), any number of language courses (0, 1, 2, 3, or 4), and any number of business courses (0, 1, 2, 3, 4, or 5).

For mathematics courses:
C(3, 0) = 1 combination
C(3, 1) = 3 combinations
C(3, 2) = 3 combinations
C(3, 3) = 1 combination

For language courses:
C(4, 0) = 1 combination
C(4, 1) = 4 combinations
C(4, 2) = 6 combinations
C(4, 3) = 4 combinations
C(4, 4) = 1 combination

For business courses:
C(5, 0) = 1 combination
C(5, 1) = 5 combinations
C(5, 2) = 10 combinations
C(5, 3) = 10 combinations
C(5, 4) = 5 combinations
C(5, 5) = 1 combination

To find the total number of different academic programs, we multiply the number of combinations for each course category together:

Total number of different academic programs = (10 * 1 * 1) + (10 * 3 * 1) + (5 * 3 * 1) + (1 * 1 * 1) + (10 * 4 * 1) + (10 * 3 * 1) + (5 * 4 * 1) + (1 * 6 * 1) + (1 * 4 * 1) + (1 * 1 * 1)
= 10 + 30 + 15 + 1 + 40 + 30 + 20 + 6 + 4 + 1
= 157

Lisa could follow 157 different academic programs.