Question

Observe the following Lists

List I	List II
A. The number of ways of answering one or more of $n$ questions	i. $\frac{{}^n{P}_r}{2r}$
B. The number of ways of answering one or more of $n$ questions is when each question has an alternative is	ii. $2^n-1$
C. The number of circular permutations of $n$ different things taken $r$ at a time is	iii. $\frac{{}^n{P}_r}{r}$
D. The number of circular permutations of $n$ things taken $r$ at a time none direction is	iv. $3^n-1$
	v. $2^n$

• A. A-ii, B-iv, C-iii, D-i
• B. A-ii, B-iii, C-i, D-iv
• C. A-iii, B-ii, C-i, D-iv
• D. A-iv, B-iii, C-ii, D-i

Answer 1

Answer: (A)

A-ii, B-iv, C-iii, D-i

A. No. of ways of answering one or more questions $={{}^{n}C}_1+{{}^{n}C}_2+{{}^{n}C}_3+{{}^{n}C}_4+...+{{}^{n}C}_n$

From binomial expansion theroem,

$2^n={{}^{n}C}_0+{{}^{n}C}_1+{{}^{n}C}_2+{{}^{n}C}_3+{n_C}_4+...+{{}^{n}C}_n$

$⟹2^n-1={{}^{n}C}_1+{{}^{n}C}_2+{{}^{n}C}_3+{{}^{n}C}_4+...+{{}^{n}C}_n$

No. of ways of answering one or more questions out of $n$ questions is $2^n-1$

Option 2

B. No. of ways of answering one question with 2 alternatives = ${{}^{n}C}_1\times 2$

No. of ways of answering two questions with 2 alternatives = ${{}^{n}C}_2\times 2^2$

.

.

No. of ways of answering r questions with 2 alternatives = ${{}^{n}C}_r\times 2^r$

Total no. of ways of answering one or more questions with $2$ alternatives $={{}^{n}C}_1\times 2^1+{{}^{n}C}_2\times 2^2+{{}^{n}C}_3\times 2^3+....+{{}^{n}C}_n\times 2^n$

From binomial expansion theorem,

$(1+2{)}^n={{}^{n}C}_0\times 2^0+{{}^{n}C}_1\times 2^1+{{}^{n}C}_2\times 2^2..+{{}^{n}C}_n\times 2^n$

So,

${}^{}{3}^n-1={{}^{n}C}_1\times 2^1+{{}^{n}C}_2\times 2^2+{{}^{n}C}_3\times 2^3+....+{{}^{n}C}_n\times 2^n$

No. of ways of answering one or more questions with 2 alternatives is $3^n-1$

C. The number of circular permutations of $n$ different things taken $r$ at a time is $\frac{{{}^{n}P}_r}{r}$

D. The number of circular permutations of $n$ different things taken $r$ at a time when order does not matter is $\frac{{{}^{n}P}_r}{2r}$