A student is allowed to choose at most n book | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the total number of ways to select at most $n$ books be $x$. Since the student must select at least one book,we have:$x = {}^{2n+1}C_1 + {}^{2

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(A) Let the total number of ways to select at most $n$ books be $x$. Since the student must select at least one book,we have:$x = {}^{2n+1}C_1 + {}^{2n+1}C_2 + \dots + {}^{2n+1}C_n = 255$We know that the sum of all combinations for $2n+1$ items is:${}^{2n+1}C_0 + {}^{2n+1}C_1 + \dots + {}^{2n+1}C_n + {}^{2n+1}C_{n+1} + \dots + {}^{2n+1}C_{2n+1} = 2^{2n+1}$Using the property ${}^{m}C_r = {}^{m}C_{m-r}$,we have ${}^{2n+1}C_0 = {}^{2n+1}C_{2n+1} = 1$ and ${}^{2n+1}C_1 = {}^{2n+1}C_{2n}$,etc.Thus,the sum can be written as:$2({}^{2n+1}C_1 + {}^{2n+1}C_2 + \dots + {}^{2n+1}C_n) + {}^{2n+1}C_0 + {}^{2n+1}C_{2n+1} = 2^{2n+1}$$2x + 1 + 1 = 2^{2n+1}$$2x + 2 = 2^{2n+1}$$x + 1 = 2^{2n}$Given $x = 255$,we get:$255 + 1 = 2^{2n}$$256 = 2^{2n}$$2^8 = 2^{2n}$$2n = 8 \implies n = 4$Therefore,the value of $n$ is $4$.

$A$ student is allowed to choose at most $n$ books from a collection of $2n+1$ books. If the total number of ways in which he can select at least one book is $255$,then the value of $n$ is:

Similar Questions

The least value of $n$ such that ${}^{(n-1)}C_6 + {}^{(n-1)}C_7 < {}^{n}C_8$ is

The least value of the natural number $n$ satisfying $C(n, 5)+C(n, 6)>C(n+1, 5)$ is

From a class of $25$ students,$10$ are to be chosen for an excursion party. There are $3$ students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

In how many ways can $10$ balls be divided between two boys,such that one receives $2$ balls and the other receives $8$ balls?

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ of $20$ distinct elements is $k$ times the number of $5$-element subsets containing $a_4$,then $k$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module