A person is permitted to select at least one | vedclass

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

Question

(A) The total number of ways to select at least one and at most $n$ coins from $(2n + 1)$ distinct coins is given by $T = {}^{2n+1}C_1 + {}^{2n+1}C_2

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(A) The total number of ways to select at least one and at most $n$ coins from $(2n + 1)$ distinct coins is given by $T = {}^{2n+1}C_1 + {}^{2n+1}C_2 + \dots + {}^{2n+1}C_n = 255$.We know that the sum of binomial coefficients is ${}^{2n+1}C_0 + {}^{2n+1}C_1 + \dots + {}^{2n+1}C_{2n+1} = 2^{2n+1}$.Since ${}^{2n+1}C_r = {}^{2n+1}C_{2n+1-r}$,we have ${}^{2n+1}C_0 = {}^{2n+1}C_{2n+1} = 1$.Also,${}^{2n+1}C_1 = {}^{2n+1}C_{2n}$,${}^{2n+1}C_2 = {}^{2n+1}C_{2n-1}$,and so on.Thus,$2({}^{2n+1}C_1 + {}^{2n+1}C_2 + \dots + {}^{2n+1}C_n) + {}^{2n+1}C_0 + {}^{2n+1}C_{2n+1} = 2^{2n+1}$.Substituting $T = 255$,we get $2(255) + 1 + 1 = 2^{2n+1}$.$510 + 2 = 2^{2n+1} \Rightarrow 512 = 2^{2n+1}$.Since $512 = 2^9$,we have $2n + 1 = 9$.$2n = 8 \Rightarrow n = 4$.

$A$ person is permitted to select at least one and at most $n$ coins from a collection of $(2n + 1)$ distinct coins. If the total number of ways in which he can select coins is $255$,then $n$ equals

Similar Questions

In how many ways can $6$ apples be distributed among $3$ boys,there being no restriction to the number of apples each boy may get?

$A$ box contains two white balls,three black balls,and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?

In how many ways can four children be made to stand in a line such that two of them,$A$ and $B$,are always together?

The number of ways of selecting $15$ teams from $15$ men and $15$ women,such that each team consists of a man and a woman,is

In how many ways can a football team of $11$ players be selected from $15$ players? In how many cases can a particular player be included?

Vedclass Test Series

Exam Paper Generator

Online Exam Module