# Factorials - Prove [15! / (5! x 10!)] + [15! / (4! x 11!)] = 16! / (5! x 11!)

## Question:

Prove [15! / (5! x 10!)] + [15! / (4! x 11!)] = 16! / (5! x 11!)

## Solution:

L.H.S.

[15! / (5! x 10!)] + [15! / (4! x 11!)]

= [(15! x 11) / (5! x 11 x 10!)] + [(15! x 5) / (5 x 4! x 11!)]

`In first term, denominator is multiplied by 11 and numerator is also multiplied by 11. In second term, denominator is multiplied by 5, numerator is also multiplied by 5. `

= [(15! x 11) / (5! x __11 x 10!__)] + [(15! x 5) / (__5 x 4!__ x 11!)]

= [(15! x 11) / (5! x 11!)] + [(15! x 5) / (5! x 11!)]

`Using the property n! = n (n - 1)!`

= [(15! x 11) + (15! x 5)] / (5! x 11!)

= [15! (11 + 5)] / (5! x 11!)

= [15! (11 + 5)] / (5! x 11!)

= (15! x 16) / (5! x 11!)

= 16 / (5! x 11!)

`Using the property n! = n (n - 1)!`

= RHS.

## Answer:

[15! / (5! x 10!)] + [15! / (4! x 11!)] = 16! / (5! x 11!) since LHS = RHS.