If binom{18}{15} + 2binom{18}{16} + binom{17} | vedclass

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

Question

(C) We use the identity $\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}$.Given expression: $\binom{18}{15} + \binom{18}{16} + \binom{18}{16} + \binom{

Vedclass · Accepted Answer

$20$

Vedclass · Answer

(C) We use the identity $\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}$.Given expression: $\binom{18}{15} + \binom{18}{16} + \binom{18}{16} + \binom{17}{16} + 1 = \binom{n}{3}$.Note that $1 = \binom{17}{17}$.So,$\binom{18}{15} + \binom{18}{16} + \binom{18}{16} + \binom{17}{16} + \binom{17}{17} = \binom{n}{3}$.Using the identity $\binom{17}{16} + \binom{17}{17} = \binom{18}{17}$,we get:$\binom{18}{15} + \binom{18}{16} + \binom{18}{16} + \binom{18}{17} = \binom{n}{3}$.Using the identity $\binom{18}{16} + \binom{18}{17} = \binom{19}{17}$,we get:$\binom{18}{15} + \binom{18}{16} + \binom{19}{17} = \binom{n}{3}$.Using the identity $\binom{18}{15} + \binom{18}{16} = \binom{19}{16}$,we get:$\binom{19}{16} + \binom{19}{17} = \binom{n}{3}$.Using the identity $\binom{19}{16} + \binom{19}{17} = \binom{20}{17}$,we get:$\binom{20}{17} = \binom{n}{3}$.Since $\binom{20}{17} = \binom{20}{20-17} = \binom{20}{3}$,we have $\binom{20}{3} = \binom{n}{3}$.Therefore,$n = 20$.

If $\binom{18}{15} + 2\binom{18}{16} + \binom{17}{16} + 1 = \binom{n}{3}$,then find the value of $n$.

Similar Questions

In an election,the number of candidates is $1$ more than the number of persons to be elected. If the voters can cast their votes in $254$ ways,find the number of candidates. ($A$ voter cannot cast more votes than the number of persons to be elected.)

There are $15$ players in a cricket team,out of which $6$ are bowlers,$7$ are batsmen,and $2$ are wicketkeepers. The number of ways a team of $11$ players can be selected from them so as to include at least $4$ bowlers,$5$ batsmen,and $1$ wicketkeeper is $.....$

$A$ box contains $2$ white balls,$3$ black balls,and $4$ red balls. In how many ways can $3$ balls be drawn from the box if at least one black ball is to be included in the draw?

In how many ways can a committee consisting of one or more members be formed out of $12$ members of the Municipal Corporation?

$A$ man has $7$ friends. In how many ways can he invite one or more of them for a tea party?

Vedclass Test Series

Exam Paper Generator

Online Exam Module