If $0 < r < s \leqslant n$and ${}^n{P_r} = {}^n{P_s}$, then the value of $r + s$ is:
$\left( A \right).$ 1
$\left( B \right).$ 2
$\left( C \right).$ $2n - 1$
$\left( D \right).$ $2n - 2$
Answer
366.6k+ views
Hint: Use formulas of permutation to find the value.
We know that:
${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}{\text{ }}\left( {{\text{Permutation Formula}}} \right)$
Given that: ${}^n{P_r} = {}^n{P_s}$
$
\therefore \dfrac{{n!}}{{\left( {n - r} \right)!}} = \dfrac{{n!}}{{\left( {n - s} \right)!}} \\
\left( {n - r} \right)! = \left( {n - s} \right)! \\
$
Also, $r < s{\text{ }}\left( {{\text{Given}}} \right)$
$\therefore - r > - s$
Adding $n$both sides, we get
$\left( {n - r} \right) > \left( {n - s} \right)$
We know that two different factorials having the same value are 0 and 1, both having factorial equal to 1.
$\therefore n - r = 1$and $n - s = 0$
$
\Rightarrow r = n - 1,s = n \\
\therefore r + s = n + n - 1 \\
r + s = 2n - 1 \\
$
Hence, the correct option is C.
Note: Whenever you see permutation, always try to expand the term by using a permutation formula which makes calculation easy.
We know that:
${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}{\text{ }}\left( {{\text{Permutation Formula}}} \right)$
Given that: ${}^n{P_r} = {}^n{P_s}$
$
\therefore \dfrac{{n!}}{{\left( {n - r} \right)!}} = \dfrac{{n!}}{{\left( {n - s} \right)!}} \\
\left( {n - r} \right)! = \left( {n - s} \right)! \\
$
Also, $r < s{\text{ }}\left( {{\text{Given}}} \right)$
$\therefore - r > - s$
Adding $n$both sides, we get
$\left( {n - r} \right) > \left( {n - s} \right)$
We know that two different factorials having the same value are 0 and 1, both having factorial equal to 1.
$\therefore n - r = 1$and $n - s = 0$
$
\Rightarrow r = n - 1,s = n \\
\therefore r + s = n + n - 1 \\
r + s = 2n - 1 \\
$
Hence, the correct option is C.
Note: Whenever you see permutation, always try to expand the term by using a permutation formula which makes calculation easy.
Last updated date: 27th Sep 2023
•
Total views: 366.6k
•
Views today: 10.66k
Recently Updated Pages
What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

What is meant by shramdaan AVoluntary contribution class 11 social science CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Is current density a scalar or a vector quantity class 12 physics JEE_Main

What is the value of 01+23+45+67++1617+1819+20 class 11 maths CBSE

How many millions make a billion class 6 maths CBSE

Draw a welllabelled diagram of a plant cell class 11 biology CBSE

Number of Prime between 1 to 100 is class 6 maths CBSE
