Question

Find the value of N ?

Answer 1

It has given that, ⁿC₀ , ⁿC₁ , ⁿC₂ , ..... ⁿCn are frequencies of (n + 1) observations 2⁰, 2¹ , 2², 2³, 2⁴ ..... 2ⁿ such that mean is 729/2ⁿ

To find : The value of n.

solution : here frequencies : ⁿC₀ , ⁿC₁ , ⁿC₂ , ..... ⁿCn

observations : 2⁰, 2¹ , 2², 2³, 2⁴ ..... 2ⁿ

mean = (x₁f₁ + x₂f₂ + x₃f₃ + ..... xnfn)/(f₁ + f₂ + f₃ + .... + fn)

⇒729/2ⁿ = (2⁰ ⁿC₀ + 2¹ × ⁿC₁ + 2² × ⁿC₂ + .... + 2ⁿ × ⁿCn)/(ⁿC₀ + ⁿC₁ + ⁿC₂ + .... + ⁿCn)

we know, (1 + x)ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x² + .... + ⁿCnxⁿ

here, ⁿC₀ + ⁿC₁ + ⁿC₂ + .... + ⁿCn = ⁿC₀ + ⁿC₁(1) + ⁿC₂(1)² + .... + ⁿCn(1)ⁿ = (1 + 1)ⁿ = 2ⁿ

similarly, (2⁰ ⁿC₀ + 2¹ × ⁿC₁ + 2² × ⁿC₂ + .... + 2ⁿ × ⁿCn) = ( 1 + 2)ⁿ = 3ⁿ

so, 729/2ⁿ = (2⁰ ⁿC₀ + 2¹ × ⁿC₁ + 2² × ⁿC₂ + .... + 2ⁿ × ⁿCn)/(ⁿC₀ + ⁿC₁ + ⁿC₂ + .... + ⁿCn) = 3ⁿ/2ⁿ

⇒729 = 3ⁿ

⇒3⁶ = 3ⁿ

⇒n = 6

Therefore the value of n is 6

Answer 2

Answer:

$\huge\boxed{\fcolorbox{white}{pink}{Answer}}$

To find n! we multiply the number from 1 to n, so to find n form n! we can start dividing that number from 2 to the number with which we get quotient as 1. let me explain with an example, let n!

______________

ɧσ℘ε ıt'ร ɧεɭ℘ʄนɭ ❤️