Pascal Triangle

Q. The number of non - negative integral solutions of $x_1+x_2+x_3+x_4\le n$ (where n is a positive integer) is:

1. ${}^{n+3}{C}_3$
2. ${}^{n+2}{C}_3$
3. ${}^{n+4}{C}_4$
4. ${}^{n+4}{C}_3$

Q. If ${}^n{C}_r=84,{}^n{C}_{r-1}=36$ and ${}^n{C}_{r+1}=126$, then the value of $n$ is

1. $9$
2. $12$
3. $11$
4. $10$

Q. The value of $\sum _{r=1}^{n}r\times r!$ is

1. $(n+1)!-1$
2. $\left(n\right)!-1$
3. $(n+1)!-n$
4. $n!$

Q. If $\frac{{}^{2n}{C}_3}{{}^n{C}_3}=11$, then the value of $n$ is

Q. If ${}^{n+1}{C}_5-{}^n{C}_4>{}^n{C}_3$, then the minimum value of $n$ is

Q. If the number of terms in $(a+b{)}^{n^2+3}$ and $(c+d{)}^{3n+4}$ are same, where $a,b,c,d\ne 0,n\in N$, then the number of possible value(s) of $n$ is

1. $0$
2. $1$
3. $2$
4. More than $2$ but finite.

Q. The number of value(s) of $r$ satisfying the equation ${}^{69}{C}_{3r-1}-{}^{69}{C}_{r^2}={}^{69}{C}_{r^2-1}-{}^{69}{C}_{3r}$ is

Q. If ${}^n{C}_4={}^n{C}_5$, then the value of ${}^n{C}_2$ is

1. $10$
2. $28$
3. $21$
4. $36$

Q. The value of $\sum _{r=1}^n\frac{r{}^n{C}_r}{{}^n{C}_{r-1}}$ is

1. $\frac{(n+1{)}^2}{2}$
2. $\frac{n^2}{2}$
3. $\frac{n(n+1)}{2}$
4. $\frac{n(n-1)}{2}$

Q. If the number of terms in the expansion of $(a+b{)}^{n^2+3}$ is $7$, then the number of value(s) of $n,(n\in N)$ is