## Additional Solved Problems

**Example 1**

A 10-digit number is formed using the digit from 0 to 9, every digit being used only once. Find the probability that the number is divisible by 4.

**Solution** The 10 digits can be arranged in 10! ways. Of these 9! will begin with the digit 0. The total number of 10 digit numbers formed is 10! − 9! = 36,28,800 − 3,62,880 = 32,65,920.

A number will be divisible by 4 if the last two digit number is divisible by 4, i.e. if it 04, 08, 12, 16, 20, 24, 28, 32, 36, 40, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 92 or 96. Of 10 digit numbers ending with

04 are 8! = 40,320

12 are 8! − 7! = 35,280 (zero is part of 8 digit numbers)

20 are 8! = 40,320

24 are 8! − 7! = 35,280 (zero is part of 8 digit numbers)

32 are 8! − ...

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