Determine the value of $A$ in the expression ${(a+b)}^n$

Question

Determine the value of $A$ in the expression below by first expressing each of the numerator and denominator in the form ${(a+b)}^n$: $$A=\frac{\displaystyle\binom 40{(2)}^4+\displaystyle\binom 41{(2)}^3(3)+\displaystyle\binom 42{(2)}^2{(3)}^2+\displaystyle\binom 43(2){(3)}^3+\displaystyle\binom 44{(3)}^4}{\displaystyle\binom 30{(4)}^3+\displaystyle\binom 31{(4)}^2+\displaystyle\binom 32(4)+\displaystyle\binom 33}$$

Answer 1

By the Binomial Theorem $(a+b)^n=\sum_{i=0}^n \binom{n}{i}a^i b^{n-i}$. The remaining task would be to determine $n,a,b$ for the expressions in the nominator and in the denominater.

Answer 2

The binomial expansion is: $$(a+b)^{n} = \sum_{k=0}^{n} \binom{n}{k} \, a^{n-k} \, b^{k} = \binom{n}{0} \, a^{n} + \binom{n}{1} \, a^{n-1} \, b + \cdots + \binom{n}{n} \, a^{0} \, b^{n}.$$

From this it can be determined that $$A = \frac{(2 + 3)^{4}}{(4 + 1)^{3}} = 5.$$