If A = {1,2,3,4,5,6} and B = {1,2,3,…,9}, then the number of strictly increasing functions from A → B is:

The correct answer is 28.

To find the number of strictly increasing functions from a domain set A to a codomain set B, we use the principles of combinatorics and functions.

Let the domain be $A=\{1,2,3,4,5,6\}$, which has $n\left(A\right)=6$ elements.
A function $f:A\to B$ is strictly increasing if for any elements $x,y\in A$ such that $x<y$, we have $f\left(x\right)<f\left(y\right)$.

Since the function is strictly increasing, the images of all elements in A must be distinct, and their order is completely determined by their values. Therefore, choosing a strictly increasing function is equivalent to selecting a subset of distinct elements from the codomain B. Once we choose $n\left(A\right)$ elements from B, there is exactly one way to arrange them in a strictly increasing order to define the function.

For a codomain set B consisting of 8 elements, the number of ways to choose 6 distinct elements is given by the combination formula:

$\mathrm{Number\; of\; functions}=\left(\genfrac{}{}{0ex}{}{8}{6}\right)$

Using the combination formula $\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$, we calculate:

$\left(\genfrac{}{}{0ex}{}{8}{6}\right)=\left(\genfrac{}{}{0ex}{}{8}{2}\right)=\frac{8\times 7}{2\times 1}=28$

Thus, the number of strictly increasing functions is 28.