### Mathematics - Binomial Theorem & Its application

#### Question - 1

For $1\le r\le n$,the value of $^{ n }{ C }_{ r }+^{ n-1 }{ C }_{ r }+^{ n-2 }{ C }_{ r }+......^{ r }{ C }_{ r },$

• A $_{ }^{ n }{ C }_{ r+1 }$
• B $_{ }^{ n +1}{ C }_{ r }$
• C $_{ }^{ n+1 }{ C }_{ r+1 }$
• D None of these

#### Question - 2

If $_{ }^{ 24}{ C }_{ 2r }=_{ }^{ 24 }{ C }_{ 2r-4 }$then r =

• A 24
• B 14
• C 10
• D 7

#### Question - 3

If $_{ }^{ n }{ C }_{ r-1 }=36,_{ }^{ n }{ C }_{ r }=84,_{ }^{ n }{ C }_{ r+1 }=126,$then

• A r=3,r=12
• B r=3,r=9
• C r=3,r=11
• D None of these

#### Question - 4

The value of the expression $^{47}C_4+\overset { 5 }{ \underset { j=1 }{ \sum { } } } ^{52-j}C_3$,is equal to

• A $^{47}C_5$
• B $^{52}C_5$
• C $^{52}C_4$
• D None of these

#### Question - 5

If n is odd natural number,the value of$\overset { 5 }{ \underset { j=1 }{ \sum { } } }\frac { { (-1) }_{ }^{ r } }{ _{ }^{ n }{ C\quad _{ r }^{ }{ } } }$is

• A 0
• B $\frac { 1 }{ n }$
• C $\frac { 1 }{ 2n }$
• D None of these

#### Question - 6

If $^{n}C_r$ denotes the number of combinations of n things taken r at a time,then the expression.$^{n}C_r+^{n}C_r-1+2\times ^{n}C_r,$equals

• A $^{n+2}C_r$
• B $^{ n+2 }{ C }_{ r+1 }$
• C $^{ n+1 }{ C }_{ r }$
• D $^{ n+1 }{ C }_{ r+1 }$

#### Question - 7

For $2\le r\le n$,$\left( \begin{matrix} n \\ r \end{matrix} \right) +2\left( \begin{matrix} n \\ r-1 \end{matrix} \right) +\left( \begin{matrix} n \\ n-2 \end{matrix} \right) =$

• A $\left( \begin{matrix} n+1 \\ r-1 \end{matrix} \right)$
• B $2\left( \begin{matrix} n+1 \\ r+1 \end{matrix} \right)$
• C $2\left( \begin{matrix} n+2 \\ r \end{matrix} \right)$
• D $\left( \begin{matrix} n+2\\ r \end{matrix} \right)$

#### Question - 8

If ${a}_n=\begin{matrix} n \\ \sum \\ r=0 \end{matrix}\frac { 1 }{ _{ }^{ n }{ C }_{ r }^{ } } \\$,then $\begin{matrix} n \\ \sum \\ r=0 \end{matrix}\frac { r }{ _{ }^{ n }{ C }_{ r }^{ } } \\$ is equal to

• A $(n-1)a_n$
• B $n a_n$
• C $\frac { n }{ 2 } { a }_{ n }^{ }$
• D None of these

The sum$\begin{matrix} m \\ \sum \\ i=0 \end{matrix}\left( \begin{matrix} 10 \\ i \end{matrix} \right) \left( \begin{matrix} 20 \\ m-i \end{matrix} \right)$, is maximum when m is $(where(\begin{matrix} p \\ q \end{matrix})=0,if\quad p • A 5 • B 10 • C 15 • D 20 #### Question - 10 The value of \(\begin{matrix} n \\ \sum \\ r=0 \end{matrix}{^{ n+r }{ C }_{ n }^{ } } \\$,equals

• A ${^{ n+m+1 }{ C }_{ n+1 }^{ } } \\$
• B ${^{ n+m+2 }{ C }_{ n}^{ } } \\$
• C ${^{ n+m+3 }{ C }_{ n-1 }^{ } } \\$
• D None of these