Question

If (n+3)! = 24 × n! then find n

Answer 1

Answer:

Put values of n one buy one and satisfy LHS =RHS

so here n=1 satisfy so ans is 1

Answer 2

Given :–

$\: \: \: \: \: \: \: \: \: \: \: \sf \: (n + 3) ! \: = 24 \times n!$

To find :–

Value of n

Solution :–

As we know that ,

$\: \: \: \: \: \: \: \: \: \boxed{ \underline{ \sf \pink{n! = n(n - 1)!}}} \: \bigstar$

Now for our convinence

$\sf \: (n+3)! \: = (n+3)(n+3-1)(n+2-1)(n+1-1)!$

$\: \: \: \sf \: (n+3)! \: = (n+3)(n + 2)(n+1)(n)!$

Now Substituting it in given equation.

$\sf \:\: (n+3)(n + 2)(n+1)(n)! = 24 \times n!$

$\sf \:\: (n+3)(n + 2)(n+1) \cancel{(n)! } = 24 \times \cancel{n!}$

$\sf \: (n + 3)(n + 2)(n + 1) = 24$

Now writing 24 as product of three consecutive numbers.

Which can be written as 4×3×2

$\sf \: (n + 3)(n + 2)(n + 1) = 4 \times 3 \times 2$

Now, Comparing L.H.S and R.H.S

$\sf \: \: n + 3 = 4$

$\sf \: n = 4 - 3$

$\: \: \: \: \: \: \boxed{ \bf \: n = 1}$

So, the value of n is 1