Question 10

**Setup:** Seven-digit numbers with digits from {0,1,2}, first digit ∈ {1,2}. Let A = numbers where 0 appears exactly twice, B = numbers where 1 appears exactly twice. **|A|:** First digit 1 or 2 (2 ways); choose 2 of the remaining 6 positions for zeros: C(6,2)=15; the other 4 positions from {1,2}: 2⁴=16. |A| = 2·15·16 = 480. **|B|:** - First digit 1: one more 1 among 6 positions (6 ways), remaining 5 positions from {0,2}: 2⁵ = 32 → 192. - First digit 2: two 1s among 6 positions: 15, remaining 4 from {0,2}: 16 → 240. |B| = 432. **|A∩B|:** - First digit 1: one more 1 in 6 ways, two 0s among remaining 5: C(5,2)=10 → 60. - First digit 2: two 1s: C(6,2)=15, two 0s among remaining 4: C(4,2)=6 → 90. |A∩B| = 150. **Answer:** 480 + 432 − 150 = 762.

No written reasoning was returned for this item.

No written reasoning was returned for this item.