There is a 7-digit telephone number with all different digits. If the digits…

When r distinct positions must each be filled by one object chosen without repetition from a pool of n available objects, and the order of placement matters, the number of ways to do so is the permutation nPr = n × (n − 1) × (n − 2) × … × (n − r + 1).

1. A telephone number here uses digit values from 0 through 9, so 10 distinct digit values are available in total.

2. The leftmost (1st) digit is fixed as 6 and the rightmost (7th) digit is fixed as 5, as given in the question. These two digit values are now used up, leaving 10 − 2 = 8 digit values still available for the rest of the number.

3. A 7-digit number has 7 positions. With the 1st and 7th positions already assigned, the 2nd through 6th positions — 5 positions in total — remain to be filled, each with one of the 8 leftover digits, with no digit repeated.

4. Filling 5 distinct positions from 8 available digits, without repetition and where order matters, is the permutation 8P5 = 8 × 7 × 6 × 5 × 4.

5. Computing the product: 8 × 7 = 56; 56 × 6 = 336; 336 × 5 = 1680; 1680 × 4 = 6720.

As a check, the factorial form gives the same result: 8P5 = 8!/(8−5)! = 8!/3! = 40320/6 = 6720 — confirming the stepwise product above.

So exactly 6720 such telephone numbers are possible, matching the option 6720.