Embibe Experts Solutions for Exercise 1: Exercise

Prove that $\frac{\left(2n\right)!}{n!}=\left\{1·3·5...\left(2n-1\right)\right\}{2}^n$.

Find the number of $4$-digit numbers that can be formed using the digits $1,2,5,6,7,$ if no two digits are the same. How many of them are divisible by $2$.

Find the number of $4$-digit numbers that can be formed using the digits $1,2,5,6,7,$ if no two digits are the same. How many of them are divisible by $25$.

Find the number of permutations of $n$ distinct things taken $r$ together, in which $3$ particular things must occur together.

If the letters of the word $BRING$ are permuted in all possible ways and the words thus formed are arranged in the dictionary order, then find the ${59}^{th}$ word.

In how many ways can $10$ persons be seated around a round a table if there are $7$ chairs? 

If ${}^{12}C_{r+1}={}^{12}C_{3r-5}$, find $r$.

Prove that for $3⩽r⩽n$.

${}^{(n-3)}C_r+3.{}^{(n-3)}C_{(r-1)}+3{.}^{(n-3)}{C}_{(r-2)}$$+{}^{(n-3)}C_{(r-3)}={}^{n}C_r$.