The thousands place cannot be 0, so it can be filled in 9 ways (digits 1 to 9). After using one digit, the hundreds place can be filled in 9 ways (including 0 now), the tens place in 8 ways and the units place in 7 ways. Total = 9 x 9 x 8 x 7 = 4536.

Choose 3 men from 7 in 7C3 ways and 2 women from 5 in 5C2 ways. 7C3 = (7 x 6 x 5)/(3 x 2 x 1) = 35 and 5C2 = (5 x 4)/(2 x 1) = 10. By the multiplication principle, total ways = 35 x 10 = 350.

total = 7C3 x 5C2

We use the property that nCa = nCb implies either a = b or a + b = n. Since r is not equal to r + 2, we must have r + (r + 2) = 10, so 2r + 2 = 10, giving 2r = 8 and r = 4.

nCa = nCb => a = b or a + b = n