Question 13 and 14, Exercise 6.3

Solutions of Question 13 and 14 of Exercise 6.3 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

In a examination, a candidate has to pass in each of $6$ subjects.
In how may ways he cannot qualify the examination?

Solution.

Candidate fails to qualify if he fails at least $1$ subject,
so we have to calculate possible ways if he fails $1$ subject, $2$
subject so on upto $6$ subjects.
Possible ways of failing $1$ subject out of $6={ }^{6} C_{1}=6$
Possible ways of failing $2$ subjects out of $6={ }^{6} C_{2}=15$
Possible ways of failing $3$ subjects out of $6={ }^{6} C_{3}=20$
Possible ways of failing $4$ subjects outsof $6={ }^{6} C_{4}=15$
Possible ways of failing $5$ subjects out of $6={ }^{6} C_{5}=6$
Possible ways of failing $6$ subjects out of $6={ }^{6} C_{6}=1$
$$\text{Total}\quad =6+15+20+15+6+1=63$$

A question papers has three parts $A$, $B$ and $C$ each containing $8$ questions.
If a student has to chose $5$ questions from $A$, and $3$ questions from $B$ and $C$.
In how many ways can he chose the questions?

Solution.

Possible ways to choose $5$ question out of $8$ in section $A$ $$ ={ }^{8} C_{5}=56 $$ Possible ways to choose $5$ question out of $\mathbf{8}$ in section $\mathrm{B} \& \mathrm{C}$ $$ ={ }^{8} C_{3}=56 $$ Total possible ways to choose question $={ }^{8} C_{5} \times{ }^{8} C_{3} \times{ }^{8} C_{3}$ $=56 \times 56 \times 56=175616$

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