If ${}^n{P_r} = {}^n{P_{r + 1}}$ and ${}^n{C_r} = {}^n{C_{r - 1}}$ , find n and r.

Question

Vedantu Content Team · Accepted Answer

Hint: By the use of Permutation and Combination formulae and properties we will find the values of n and r.

Given,
${}^n{P_r} = {}^n{P_{r + 1}} \to (1)$
Since, we know that the number of permutations of n objects taken r at a time is ${}^n{P_r}$ and its value is $\dfrac{{n!}}{{(n - r)!}}$ . Similarly, the value of ${}^n{P_{r + 1}}$ is equal to $\dfrac{{n!}}{{(n - r - 1)!}}$. So, applying the formulae in equation (1), we get
$
   \Rightarrow \dfrac{{n!}}{{(n - r)!}} = \dfrac{{n!}}{{(n - r - 1)!}} \\
   \Rightarrow (n - r - 1)! = (n - r)! \to (2) \\
$
As, we know that $n! = n*(n - 1)!$ . Therefore, $(n - r)! = (n - r)*(n - r - 1)!$ . Rewriting equation (2), we get
$ \Rightarrow (n - r - 1)! = (n - r)*(n - r - 1)!$
The term $(n - r - 1)!$ gets cancelled on both sides, then the equation will be
$
   \Rightarrow 1 = (n - r) \\
   \Rightarrow n = r + 1 \to (3) \\
$
Now, let us consider the given condition ${}^n{C_r} = {}^n{C_{r - 1}} \to (4)$. As, we know the property of combination i.e.., if ${}^n{C_x} = {}^n{C_y}$ then $x = y$ or $x + y = n$. Therefore, using the property of combinations, equation (4) can be written as
$
   \Rightarrow r + r - 1 = n \\
   \Rightarrow 2r - 1 = n \\
$
Let us substitute the value of n from equation (3) in the above equation, we get
$
   \Rightarrow 2r - 1 = r + 1 \\
   \Rightarrow 2r - r = 1 + 1 \\
   \Rightarrow r = 2 \\
$
Hence, the obtained value of r is 2.let us substitute the value of r in equation (3), we get
$
   \Rightarrow n = 2 + 1 \\
   \Rightarrow n = 3 \\
$
Therefore, the obtained value of n is 3 and r is 2.

Note: Here, we have considered $r + r - 1 = n$ condition, because if we consider the condition $r = r - 1$ the value of r can’t be computed.

If ${}^n{P_r} = {}^n{P_{r + 1}}$ and ${}^n{C_r} = {}^n{C_{r - 1}}$ , find n and r.

NCERT Book Solutions

Reference Book Solutions

Question Papers

Exams

Quick Links

If ${}^n{P_r} = {}^n{P_{r + 1}}$ and ${}^n{C_r} = {}^n{C_{r - 1}}$ , find n and r.

Book your FREE Online counselling session

Your FREE Online counselling session has been booked!

Please try again later

NCERT Book Solutions

Reference Book Solutions

Question Papers

Exams

Quick Links