Pascal Triangle

Q.

What is formula for ${\left(x-y-z\right)}^3$

Q. If ${}^n{C}_4,{}^n{C}_5,{}^n{C}_6$ are in A.P., then the value(s) of $n$ is/are

1. $7$
2. $14$
3. $8$
4. $16$

Q. If ${}^{n+1}{C}_3=2{}^n{C}_2$, then the value of $n$ is

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. $\text{If}\frac{{}^{n+2}{C}_6}{{}^{n-2}{P}_2}=11$, then $n$ satisfies the equation:

1. $n^2+2n-80=0$
2. $n^2+5n-84=0$
3. $n^2+3n-108=0$
4. $n^2+n-110=0$

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

Q. The value of ${}^{19}{C}_2$ is

1. $171$
2. $342$
3. $\frac{19!}{2!}$
4. $\frac{19!}{17!}$

Q. If $(n+1)!=12\times (n-1)!$, then the value of $n$ can be

1. $-4$
2. $3$
3. $-3$
4. $4$

Q. If ${}^n{C}_4,{}^n{C}_5,{}^n{C}_6$ are in A.P., then the value(s) of $n$ is/are

1. $7$
2. $14$
3. $8$
4. $16$

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. Let $f:(0,\infty )\to R$ be given by
$f\left(x\right)=\underset{1/x}{\overset{x}{\int }}{e}^{-(t+\frac{1}{t})}\frac{dt}{t}.$
Then

1. $f\left(x\right)$ is monotonically increasing on $[1,\infty )$
2. $f\left(x\right)$ is monotonically decreasing on $(0,1)$
3. $f\left(x\right)+f\left(\frac{1}{x}\right)=0,$ for all $x\in (0,\infty )$
4. $f\left(2^x\right)$ is an odd function of $x$ on $R$

Q. If ${}^n{C}_5={}^n{C}_{10}$, then the value of ${}^{16}{C}_n$ is

1. $1$
2. $16$
3. $15$
4. $15!$

Q. If ${}^{18}{C}_{2r}={}^{18}{C}_{r+3}$, then the value of $r$ is/are

1. $2$
2. $3$
3. $4$
4. $5$

Q. If ${}^{18}{C}_{2r}={}^{18}{C}_{r+3}$, then the value of $r$ is/are

1. $2$
2. $3$
3. $4$
4. $5$

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. 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. 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

Q.

Find the value of r, if ${}^{20}{C}_{r+1}={}^{20}{C}_{3r-1}$

1. 1

2. 2

3. 4

4. 5

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

1. $2{}^{m+1}{C}_2$
2. $3{}^{m+1}{C}_4$
3. ${}^{m+1}{C}_4$
4. $3{}^{m+1}{C}_2$

Q. If ${}^n{C}_2+{}^n{C}_3={}^{n+1}{C}_r$, then the minimum possible value of $r$ is

1. $0$
2. $3$
3. $2$
4. $1$

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 $\sum _{r=0}^4\frac{{}^4{C}_r}{{}^4{C}_r+{}^4{C}_{r+1}}$ is

1. $\frac{1}{2}$
2. $3$
3. $\frac{1}{3}$
4. $2$

Q. Compute $\frac{8!}{6!\times 2!}$

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

Q. The expression ${}^n{C}_r+2{}^n{C}_{r-1}+{}^n{C}_{r-2}$ is equal to

1. ${}^{n+1}{C}_r$
2. ${}^{n+2}{C}_{r+1}$
3. ${}^{n+2}{C}_r$
4. ${}^{n+1}{C}_{r+1}$

Q. If ${}^n{C}_5={}^n{C}_{10}$, then the value of ${}^{16}{C}_n$ is

1. $1$
2. $15$
3. $16$
4. $15!$

Q. If ${}^n{P}_r{=}^n{P}_{r+1}$ and ${}^n{C}_r{=}^n{C}_{r-1}$, then the values of $n$ and $r$ are

1. $r,3$
2. $3,2$
3. $4,2$
4. $3,4$

Q. The number of words that can be formed by the letters of $WORDS$ is

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.

Let a relation R be defined by R = {(4, 6); (1, 4); (4, 6); (7, 6); (3, 7)} then $R^{-1}$ oR is

1. {(1, 1), (4, 4), (4, 7), (7, 4), (7, 7), (3, 3)}

2. {(1, 1), (4, 4), (7, 7), (3, 3)}

3. {(1, 5), (1, 6), (3, 6)}

4. None of these