(i) To find the number of different numbers without restrictions, we calculate the permutations of the digits 223 677 888. The total number of digits is 9, with repetitions: two 2s, two 3s, and three 8s. The formula for permutations with repetitions is:
\(\frac{9!}{2!2!3!} = \frac{362880}{8 \times 2 \times 6} = 15120 \text{ ways}\)
(ii) To find the number of even numbers, the last digit must be 2, 6, or 8. We calculate the permutations for each case:
- Ending with 2: Arrange the remaining 8 digits (23677888), with repetitions of two 8s and two 7s:
\(\frac{8!}{2!2!} = \frac{40320}{4 \times 2} = 3360 \text{ ways}\)
- Ending with 6: Arrange the remaining 8 digits (22377888), with repetitions of two 2s, two 8s, and two 7s:
\(\frac{8!}{2!2!2!} = \frac{40320}{8} = 1680 \text{ ways}\)
- Ending with 8: Arrange the remaining 8 digits (22367788), with repetitions of two 2s and two 7s:
\(\frac{8!}{2!2!} = \frac{40320}{4 \times 2} = 5040 \text{ ways}\)
Total even numbers:
\(3360 + 1680 + 5040 = 10080 \text{ ways}\)
