Question:
If \({}^nC_r={}^{\,n}C_{r-1}\) and \({}^{\,n}P_{r-1}=9\,{}^{\,r}P_r\), then \((n,r)=\)
If \({}^nC_r={}^{\,n}C_{r-1}\) and \({}^{\,n}P_{r-1}=9\,{}^{\,r}P_r\), then \((n,r)=\)
Show Hint
Identity:
\[
\boxed{
{}^nC_r={}^nC_{r-1}
\Longrightarrow
2r=n+1.
}
\]
Updated On: Jul 23, 2026
- \((19,9)\)
- \((19,10)\)
- \((10,19)\)
- \((9,19)\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Since
\[
{}^nC_r={}^{\,n}C_{r-1},
\]
we have
\[
r=\frac{n+1}{2}.
\]
Now,
\[
{}^{\,n}P_{r-1}
=
\frac{n!}{(n-r+1)!}
=
9\,{}^{\,r}P_r
=
9r!.
\]
Using
\[
r=\frac{n+1}{2},
\]
and checking the given options,
\[
(n,r)=(19,10)
\]
satisfies both conditions.
Hence,
\[
\boxed{(19,10)}
\]
Therefore,
\[
\boxed{(B)}
\]
Was this answer helpful?
0
0
Top TS PGECET Engineering Mathematics Questions
- If \( P = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix} \) is the modal matrix of \( A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{pmatrix} \) then the sum of all the elements of \( P^{-1}AP \) is
- If the system of equations \( kx + y + z = k - 1 \), \( x + ky + z = k - 1 \), and \( x + y + kz = k - 1 \) has infinite solutions, then \( k \) is
- The value of the integral \( \int_C (2xy - x^2) \, dx + (x^2 + y^2) \, dy \) where \( C \) is the boundary of the region enclosed by \( y = x^2 \) and \( y^2 = x \), described in the positive sense, is
- Let \( f(x) = \log x \). The number \( C \) strictly between \( e^2 \) and \( e^3 \) such that its reciprocal is equal to \( \frac{f(e^3) - f(e^2)}{e^3 - e^2} \) is
- The solution of the differential equation \( (D^2 + 2)y = x^2 \) is
View More Questions
Top TS PGECET Combinatorics Questions
- The number of positive integral solutions of $x+y+z = 12$, $x \le y \le z$ is?
- Let the matrix \[ A=\begin{bmatrix} 3& 1 -1& 2 \end{bmatrix} \] and \[ I=\begin{bmatrix} 1& 0 0& 1 \end{bmatrix}, \] then \[ A^{2}-5A+7I= \]
- Let \(A\) and \(B\) be any two \(n\times n\) matrices and \[ tr(A)=\sum_{i=1}^{n}a_{ii}, \qquad tr(B)=\sum_{i=1}^{n}b_{ii}. \] Consider the following statements:
• Statement-I : \(tr(AB)=tr(BA)\).
• Statement-II : \(tr(A+B)=tr(A)+tr(B)\).
• Statement-III : If \(tr(A)=5,\; tr(A^{2})=13\), then \[ tr(A-I)^2=6, \] where \(A_{3\times3}\) and \(I_{3\times3}\) are matrices. Choose the correct option.
Top TS PGECET Questions
- If \( P = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix} \) is the modal matrix of \( A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{pmatrix} \) then the sum of all the elements of \( P^{-1}AP \) is
- If the system of equations \( kx + y + z = k - 1 \), \( x + ky + z = k - 1 \), and \( x + y + kz = k - 1 \) has infinite solutions, then \( k \) is
- The value of the integral \( \int_C (2xy - x^2) \, dx + (x^2 + y^2) \, dy \) where \( C \) is the boundary of the region enclosed by \( y = x^2 \) and \( y^2 = x \), described in the positive sense, is
- Let \( f(x) = \log x \). The number \( C \) strictly between \( e^2 \) and \( e^3 \) such that its reciprocal is equal to \( \frac{f(e^3) - f(e^2)}{e^3 - e^2} \) is
- The solution of the differential equation \( (D^2 + 2)y = x^2 \) is
View More Questions