Problem 11 · Count by blocks and gaps · 3 points
Made by AI and not reviewed by a real person.
Fourteen lily pads sit in a row. A frog paints exactly seven gold, forming exactly three groups of consecutive gold pads, each with at least two pads. How many paintings are possible?
- 168
- 105
- 56
- 360
- 120
Show a hint
- Think about the lengths of the gold groups, then about the blank spaces around and between them.
- Two blank pads are needed to separate the three gold groups. How can the remaining blanks be shared among the four spaces before, between, and after the groups?
Show the answer and solution
Answer: A — 168
The gold groups must have lengths 3, 2, and 2, in any of three orders. Of the seven blank pads, reserve one for each gap between groups. The other five can be distributed among the four spaces before, between, and after the groups in C(8,3) = 56 ways. Thus there are 3 × 56 = 168 paintings. The answer 56 misses the three possible positions of the longer gold group.