If ${}^n{P}_r=n(n-1)(n-2)\cdots (n-r+1)$ and ${}^n{P}_3=24$, then the value of $n$ will be:

The definition of permutation ${}^n{P}_3$ is the product of $3$ consecutive decreasing integers starting from $n$:
$${}^n{P}_3=n(n-1)(n-2)$$

We are given that this value equals $24$:

$$n(n-1)(n-2)=24$$

Let us find three consecutive integers whose product is $24$.
By factoring $24$, we see:

$$24=4\times 6=4\times 3\times 2$$

Arranging them in descending order:

$$24=4\times 3\times 2$$

Comparing this with $n(n-1)(n-2)$:

$$n=4$$