Mathematics - Binomial Theorem & Its application

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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

Question - 9

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
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