Question

A university student is selecting courses for his next semester. He can choose from 3 humanities courses and 10 science courses. In how many ways can he choose 7 courses if fewer than 2 must be humanities courses?

Answer 1

The university student can choose 7 courses in a total number of ways calculated by adding the combinations of 0 humanities and 7 science courses (10C7) to the product of the combination of 1 humanities course and 6 science courses (3C1 × 10C6).

Step-by-step explanation:

The question is asking to determine in how many ways a university student can choose 7 courses, given that fewer than 2 must be humanities courses. There are 3 humanities courses and 10 science courses to choose from.

To solve this, we consider the two possible scenarios that satisfy the condition of fewer than 2 humanities courses:

1. The student chooses 0 humanities courses and 7 science courses.
2. The student chooses 1 humanities course and 6 science courses.

We will calculate the possibilities for each scenario and then sum them for the total number of ways.

Scenario 1:

Choosing 0 humanities courses and 7 science courses:
The number of ways to choose 7 from 10 science courses is a combination given by 10C7 (10 choose 7).

Scenario 2:

Choosing 1 humanities course and 6 science courses:
The number of ways to choose 1 from 3 humanities courses is 3C1 (3 choose 1), and the number of ways to choose 6 from 10 science courses is 10C6 (10 choose 6).

The total ways to choose the courses is the sum of the ways from both scenarios:

Total = 10C7 + (3C1 × 10C6)

This sum gives us the total number of ways the student can choose their courses under the given conditions.