If the total number of combinations of $n$ different things taken one or more at a time is $127$ , then $n$ $=$

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Solution

The correct option is B $7$
The total number of combinations of $n$ different things taken one or more at a time $=^{n} C_{1} +^{n} C_{2} +^{n} C_{3} . . . . . . . . . . . . . .^{n} C_{n}$
We know that $^{n} C_{0} +^{n} C_{1} +^{n} C_{2} +^{n} C_{3} . . . . . . . . . . . . . .^{n} C_{n} = 2^{n}^{n} C_{1} +^{n} C_{2} +^{n} C_{3} . . . . . . . . . . . . . .^{n} C_{n} = 2^{n} - 1$
Given the total combinations is $= 127 2^{n} - 1 = 127 2^{n} = 2^{7} n = 7$

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List I	List II
A. The number of ways of answering one or more of $n$ questions	i. $\frac{^{n} P_{r}}{2 r}$
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. $2^{n}$

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