Question

A student is to answer 8 out of 10 questions on an exam.

(i) How many choices has (s)he?

2. How many choices if (s)he must answer the first 3 questions?
3. How many if (s)he must answer at least 4 of the first five questions?

Answer 1

A student is to answer 8 out of 10 questions on an exam.

Total number of questions n = 10

1. You have to randomly choose 8 questions from 10.

So you can solve it by using combination

ⁿC_r = ¹⁰C₈ = 10! /8!*2! = 10*9/2 = 45

So, he or she has 45 choices.

2. Here, you have been asked to answer the first 3 questions without choosing.

So, after first 3 questions, the remaining 5 questions out of 7 questions he or she has to solve.

That means how many ways he or she can choose remaining 5 questions out of 7 questions

ⁿC_r = ⁷C₅ = 7! /5!*2! = 7*6/2 = 21

There are 21 ways (s)he must answer first 3 questions.

3. How many if (s)he must answer at least 4 of the first five questions?

Here, (s)he must answer atleast 4 of the first five questions means first 4 questions or first 5 questions.

Suppose (s)he must answer first 4 answers out of first 5 questions and remaining 4 questions out of 5 (s)he has to solve.

⁵C₄ * ⁵C₄   = 5! 5!/4!4!*1!1! = 5*5 = 25

Suppose (s)he must answer first 5 answers without choosing

and there are 3 questions out of 5 (s)he has to solve.

ⁿC_r = ⁵C₅⁵C₃ = 1 * 5! /3!*2! = 5*4/2 = 10

Total number of ways = 25 + 10 = 35

35 ways if (s)he must answer atleast 4 of the first five questions.