Question

gys please see attached image and tell me where is the mistake


because as I solved answer comes (n-3)(n-4)=42​

Answer 1

Solution :-

Now before solving :-

→ ⁿPₘ = (n!)/(n -m)!

→ It is the number of ways in which m out of n objects can be permuted (arranged) .

• n! = n (n-1) (n -2) .... (3)(2)(1)

Now the question :-

${}^nP_5 = 42 \: {}^nP_3$

$\implies \dfrac{n!}{(n-5)!} = 42 \times \dfrac{n!}{(n-3)!}$

$\implies \dfrac{\cancel{n!}}{(n-5)!} = 42 \times \dfrac{\cancel{n!}}{(n-3)!}$

$\implies \dfrac{1}{(n-5)!} = 42 \times \dfrac{1}{(n-3)!}$

$\implies (n-3)! = 42 \times (n-5)!$

$\small{ \implies (n-3)(n-4) \times (n-5)! = 42 \times (n-5)!}$

$\small{ \implies (n-3)(n-4) \times \cancel{ (n-5)! }= 42 \times \cancel{(n-5)!} }$

$\implies (n-3)(n-4) = 42$

$\implies n^2 - 7n + 12 = 42$

$\implies n^2 - 7n - 30 = 0$

$\implies n^2 + 3n - 10n - 30 = 0$

$\implies n(n +3) -10(n + 3) = 0$

$\implies (n -10)(n +3) = 0$

$\implies n = -3 \: or \: 10$

But as n cannot be negative.

Hence n = 10

Answer 2

P(n,5) = n!/(n-5)! = n(n - 1)(n - 2)(n - 3)(n - 4)(n - 5)!/((n - 5)!

P(n,3) = n!/(n - 3)! = n(n - 1)(n - 2)(n - 3)!/(n - 3)!

Multiply this by 42, then it will be :

n * (n - 1) * (n - 2) * (n - 3) * (n - 4) = 42 * n * (n - 1) * (n - 2)

(n - 3)(n - 4) = 42

n^2 - 7n + 12 = 42

n^2 - 7n -30 = 0

(n - 10)(n + 3) = 0

n = 10, -3

Therefore, the value of n = 10.