If the number 7875 can be prime factorised as $7875 = 3^{\mathrm{a}} \times {\mathrm{b}}^3 \times {\mathrm{c}}^1$, then what are the respective values of a, b, and c? 

By the fundamental theorem of arithmetic, it is known that every composite number can be factorised as the product of primes in a unique way. Therefore, the number 7875 can be prime factorised as

$7875 = 3 \times 3 \times 5 \times 5 \times 5 \times 7$

$\Rightarrow 7875 = 3^2 \times 5^3 \times 7^1$

On comparing it with $7875 = 3^{\mathrm{a}} \times {\mathrm{b}}^3 \times {\mathrm{c}}^1$, we obtain a = 2, b = 5, and c = 7

Thus, the respective values of a, b, and c are 2, 5, and 7.