Permutations and Combinations

Q. If ${}^5{C}_r=r\cdot {}^5{C}_{r-1}$, then the number of value(s) of $r$ is

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

Q. Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that $Y\subseteq X,Z\subseteq XandY\cap Z$ is empty is:

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

Q. Let $f$ be the subset of $Z\times Z$ defined by $f=\left\{\right(ab,a+b):a,b\in Z\}$. Is $f$ a function from $Z$ to $Z$ : justify your answer

Q. The number of terms in the expansion of $(4x^2-16x+16{)}^5$ is

Q. Which of the following(s) pair have same number of terms $(x,y\in R)$

1. $(x+y{)}^9-2,(2x-3y{)}^{10}$
2. $1-(2+x{)}^m,(x^2+5{)}^m,$ where $m\in N$
3. $(x^2-2x+1{)}^5,(x^2+1{)}^{10}$
4. $(x^2-2x+1{)}^5,(x^2+1{)}^{11}$

Q. If set $A=\left\{\right(r,s)|r<s;r,s\in W\}$, then the number of elements in set $A$ such that ${}^7{C}_r+{}^7{C}_{r-1}={}^8{C}_s$ is

1. $8$
2. $6$
3. $4$
4. $3$