Properties of Binomial Coefficients

If ${}^{\mathrm{15}}{C}_{3r}={}^{15}{C}_{r+3}$, then the value of $r$ is

If  ${{}^{20}C}_{r+2}={{}^{20}C}_{2r-3}$, find ${{}^{12}C}_r$.

If ${}^{45}C_{9a}={}^{45}C_{3a+9}$ and ${}^{63}C_{9b}={}^{63}C_{3b+3}$, then what should come in the place of $\left(?\right)$ for the expression given below?

$a+b=?$

If ${}^{13}C_{2x}+{}^{13}C_{2x-1}={}^{14}C_{2y}$, then find the value of $x+y$.

If $\sum _{i=1}^n{\left(\frac{{}^{n}C_{i-1}}{{}^{n}C_i+{}^{n}C_{i-1}}\right)}^3=\frac{36}{13}$ then $n$ is equal to

Let $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_n{x}^n,$ where $C_r={}^{n}C_r$ and $\left(C_0+C_1\right)\left(C_1+C_2\right)\dots \left(C_{n-1}+C_n\right)=A{C}_1{C}_2\dots C_n$, then for $n=5, A$ is equal to

The value of the expression ${}^{n}C_{r+1}+{}^{n}C_{r-1}+2 {}^{n}C_r$ is, where ${}^{n}C_r$ is the number of selection of $r$ things out of $n$ things

If ${}^{20}C_r{=}^{20}{C}_{r-10}$, then ${}^{18}C_r$ is equal to

Prove that 

${}^{45}\mathrm{C}_8+\underset{r=1}{\overset{7}{\sum {}^{52-r}\mathrm{C}_7}}+\sum _{k=1}^5{}^{57-k}\mathrm{C}_{50-k}={}^{57}\mathrm{C}_8$

Prove that, ${}^n\mathrm{C}_r+2.{}^n\mathrm{C}_{r-1}+{}^n\mathrm{C}_{r-2}={}^{n+2}\mathrm{C}_r$

If ${}^n\mathrm{C}_{12}={}^n\mathrm{C}_8$, find ${}^n\mathrm{C}_{17}$ and ${}^{22}\mathrm{C}_n$.

Evaluate:

${}^{47}\mathrm{C}_4+\underset{j=0}{\overset{3}{{\sum }^{50-j}{\mathrm{C}}_3}}$

Evaluate:

${}^{47}\mathrm{C}_4+\underset{r=1}{\overset{5}{{\sum }^{52-r}{\mathrm{C}}_3}}$

If ${}^{25}\mathrm{C}_r={}^{25}\mathrm{C}_{2\mathrm{r}+1}$, find the value of ${}^r\mathrm{C}_5$.

How many $4-$4-digited numbers can be formed using $1, 2, 3, 4, 5$ without repetition, such that they are divisible by $4$?

If ${}^{n}C_{r-1}=36,{}^n C_r=84$ and ${}^{n}C_{r+1}=126,$ find the values of $n$ and $r$. 

Prove that ${}^{4n}C_{2n}:{}^{2n}C_n=[1.3.5\dots (4n-1\left)\right]:[1.3.5\dots .(2n-1){]}^2$.

Find the value of ${}^{15}C_8+{}^{15}C_9-{}^{15}C_6-{}^{15}C_7$

${}^{n}P_r={}^{n}P_{r+1}$ and ${}^{n}C_r={}^{n}C_{r-1}$ find the value of $n$ and $r$.

$\left|\left[\sin x\right]\right|+{}^{\lambda -1}C_5={}^{\lambda }C_6$, where $\left[*\right]$ denotes greatest integer function and $x$ is odd multiple of $\frac{\mathrm{\pi }}{2}$, then the value of $\lambda$ is