Factorials!

Number Theory Level 2

Solve for x x : 7 ! 2 ! ( 3 ! − 1 ) = ( 3 x ) ! ( x + 3 ) ! \large \dfrac{7!}{2! (3! - 1)} = \dfrac{(3x)!}{(x+3)!}

Notation : ! ! denotes the factorial function. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1\times2\times3\times\cdots\times8 .


The answer is 3.

Abhiram Rao by Abhiram Rao , 18, India

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1 solution

Sabhrant Sachan
May 12, 2016

I am going to solve L.H.S, And then by Comparison i will Find the value of x

⟹ 7 ! 2 ! ( 3 ! − 1 ) ⟹ 7 ! 10 ⟹ 7 ! 10 × 8 × 9 8 × 9 ⟹ 9 ! 10 × 9 × 8 ⟹ 9 ! 6 × 5 × 4 × 3 × 2 × 1 ⟹ 9 ! 6 ! ⟹ ( 3 × 3 ) ! ( 3 + 3 ) ! Answer = 3 \implies \dfrac{7!}{2!(3!-1)} \implies \dfrac{7!}{10} \\ \implies \dfrac{7!}{10}\times\dfrac{8\times9}{8\times9} \implies \dfrac{9!}{10\times9\times8} \\ \implies \dfrac{9!}{6\times5\times4\times3\times2\times1} \implies \dfrac{9!}{6!} \\ \implies\dfrac{(3\times3)!}{(3+3)!} \\ \text{Answer =} \boxed{3 }

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