Let $a$ be the number of ways in which $C10$ the cycle graph with $10$ vertices can be coloured with $11$ colours and let $b$ be the number of ways $K11$ , the complete graph with $11$ vertices, can be coloured with $20$ colours.
(a) Write $a$ and $b$ as the product of powers of prime numbers. Hint: there is just a single prime factor of a that is greater than $25$.
(b) Find the greatest common divisor of $a$ and $b$.
(c) Find the lowest common multiple of a and b.
Hint: The Chromatic polynomial $P(x)$ of $K_n$ is known. Evaluate it at $20$. The factorization comes easily from the formula.
If you don't know yet the Chromatic polynomial of $C_{10}$ yet you can calculate it by deletion-contraction. For the factorization, the hint makes it easy: find all primes less than 25, and their powers in this number.