Let $Y=\{1,2,3,\dots ,100\}$ and let $h:Y\to Y$ be a strictly increasing function. The total number of functions $g:Y\to Y$ satisfying $g(h(j))=h(g(j))$ for all $j\in Y\; are\; ?$​

Correct option is C

Let $Y=\{1,2,3,\dots ,100\}$ and let $h:Y\to Y$ be a strictly increasing function.

Since h is strictly increasing, h(1) <h(2) < ⋯ <h(100). This means that h is injective.

Also, since h:Y→Y, we have 1≤ h(j) ≤ 100 for all j∈Y.

Let h(Y)={h(1),h(2),…,h(100)}. Since h is strictly increasing, h(Y) has 100 distinct elements.

Thus, h(Y)=Y. This means h is a bijection, and thus a permutation of Y.

Since h is a strictly increasing function from Y to Y, we must have h(j)=j for all j∈Y.

This is because if h(1)>1, then h(2)>h(1)>1, and eventually h(100)>100, which contradicts h(100)∈Y.

Similarly, if h(1)<1, it is impossible.

Therefore, h(j)=j for all j∈Y, which means h is the identity function.

The given condition is g(h(j))=h(g(j)) for all j∈Y.

Since h(j)=j, we have g(j)=g(h(j))=h(g(j))=g(j).

Thus, the condition becomes g(j)=g(j), which is always true.

This means that g can be any function from Y to Y.

The number of functions $g:Y\to Y$ is 

​$$100^{100}\\\implies \textbf{Option C is correct}$$​​