Mr. and Mrs. smith had invited 9 of their friends and their spouses for party…
2025
Mr. and Mrs. smith had invited 9 of their friends and their spouses for party at wiki beach-resort (The stand for group photograph), if Mr. smith never stand next to Mrs. smith, then in how many ways the group can be arranged in a row.
- A.
20!
- B.
19! +18!
- C.
18* 19!
- D.
2*19!
Attempted by 318 students.
Show answer & explanation
Correct answer: C
There are 20 people in total: Mr. and Mrs. Smith plus 9 friends and their spouses.
Method 1 (complementary counting):
Total arrangements without restriction = 20!.
Arrangements where Mr. and Mrs. Smith are together: treat them as a single block → 19! arrangements for the blocks, and 2 internal orders for the couple, giving 2 × 19!.
So arrangements where they are not together = 20! − 2 × 19! = 19! × (20 − 2) = 18 × 19!.
Method 2 (gap method, alternative):
Arrange the other 18 people in 18! ways. This creates 19 gaps (including the ends).
Place Mr. and Mrs. Smith into two different gaps in order: 19 × 18 choices.
Total = 18! × 19 × 18 = 18 × 19!.
Final answer: 18 × 19!.