The 26 alphabets expansion

"Proof of Formula"

$142506 = 2 \times 3^3 \times 7 \times 13 \times 29$

$\begin{aligned} \dbinom{N+26-1}{26-1} & = 142506 \\ \dbinom{N+25}{25} & = 142506 \end{aligned}$

It is quite interesting to note that $N<6$ , because $142506$ is not a multiple of $31$ . With these many terms i thought the

answer will be in atleast $2$ digits . If you argue about taking a $N$ bigger than $31$ then $142506$ is not divisible by $32+9=41$

which is another prime number , and i can go on like this . So we just need to find $\dbinom{N+25}{25}$ for $N = 5,4,3$ . You will get $N=5$ as the answer .