Notations of Combinations

Q. The correct expression of ${}^8{\text{C}}_4$ is

1. $\frac{8\times 7\times 6\times 5\times 4}{1\times 2\times 3\times 4}$
2. $\frac{8\times 7\times 6\times 5}{1\times 2\times 3\times 4}$
3. $\frac{8\times 7\times 6\times 5\times 4}{1\times 2\times 3\times 4!}$
4. $\frac{8\times 7\times 6\times 5!}{1\times 2\times 3\times 4!}$

Q. What is the value of ${}^{10}{C}_8+{}^{10}{C}_2$?

1. $45$
2. $100$
3. $90$
4. $50$

Q. What is the value of ${}_7{\text{C}}_2?$

1. $21$
2. $18$
3. $14$
4. $49$

Q.

Write the first three terms of the sequence $a_n=3n^2-2$.

Q.

If ${(666...ntimes)}^2+(8888...ntimes)=(4444...Ktimes)$then $K$ is

Q. Which of the following is correct number of ways for selecting $n$ different things taken $k$ at a time?

1. $\frac{n!}{(n-k)!}$
2. $\frac{n!}{(n-k)!\times k!}$
3. ${}^n{C}_k$
4. $\frac{{}^n{P}_k}{k!}$

Q. Which of the following are correct representation of ${}^n{C}_r$?

1. ${}_n{C}_r$
2. $C_r^n$
3. $\left({}_r^n\right)$
4. None of the above.

Q. If ${}_n{C}_r=60$ and ${}^n{P}_r=360$, then find $r$?

1. $5$
2. $3$
3. $6$
4. Can't be determined

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

Q. ${}^n{\text{C}}_r=\frac{n!}{(n-r)!\times r!}$ is applicable only when

1. 
   $r\le n$
2. 
   $0\le r\le n$
3. 
   $0<r\le n$
4. 
   All of the above.

Q.

Write the first four terms in the list of numbers defined by $a_n=\frac{{\left(n-1\right)}^2}{3}$

Q.

Write the first four terms in the list of numbers defined by $a_n={\left(-1\right)}^{n-1}{n}^4$