How many distinct 5-digit positive integers can be formed using the digits $1, 2, 3, 4, 5, 6, 7$ if **no digit is repeated** within an integer?

This is a permutation: select **and order** $5$ digits from a pool of $7$. Number of choices per slot: - 1st digit: $7$ - 2nd digit: $6$ (one already used) - 3rd digit: $5$ - 4th digit: $4$ - 5th digit: $3$ Product: $7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 = 2{,}520$. Equivalently, $\dfrac{7!}{(7-5)!} = \dfrac{7!}{2!} = 2520$. - **A** ($5! = 120$) would be the count if there were only 5 digits to permute among 5 positions — wrong pool size. - **B** ($7^{5}$) allows repetition, which the problem forbids. - **C** ($7!$) orders **all seven** digits, but we are choosing only 5.