One writes all numbers from 50 to 99 without the digits 2 and 7. How many…

Concept

When a number is built place by place, the fundamental counting principle applies: if one place can be filled in m ways and a second place can be filled in n ways independently of the first, the two places together can be filled in m × n ways. A digit that is barred is barred in every place where it could stand, so each place is counted with its own reduced set of allowed digits.

Application

1. Every number from 50 to 99 is a two-digit number, so it has a tens place and a units place.

2. Tens place: within this range it can be 5, 6, 7, 8 or 9 only. The digit 7 is barred, leaving 4 allowed digits: 5, 6, 8 and 9.

3. Units place: any digit from 0 to 9 may stand here. The digits 2 and 7 are barred, leaving 8 allowed digits: 0, 1, 3, 4, 5, 6, 8 and 9.

4. The two places are filled independently of each other, so the count of numbers written is 4 × 8 = 32.

Cross-check

Counting by removal instead of by construction gives the same total.

• The range 50 to 99 contains 50 numbers in all.

• Numbers carrying a 7: the whole decade 70 to 79 (10 numbers) together with 57, 67, 87 and 97 (4 numbers), so 14 in all.

• Numbers carrying a 2: 52, 62, 72, 82 and 92, so 5 in all.

• The number 72 stands in both lists, so the count of barred numbers is 14 + 5 - 1 = 18.

• Numbers written: 50 - 18 = 32, matching the product above.

Result

32 numbers are written.