In how many ways can 15 indistinguishable fish be placed into 5 different…
2015
In how many ways can 15 indistinguishable fish be placed into 5 different ponds, so that each pond contains at least one fish ?
- A.
1001
- B.
3876
- C.
775
- D.
200
Attempted by 38 students.
Show answer & explanation
Correct answer: A
Key idea: convert the problem to counting positive integer solutions and use stars and bars.
Step 1: Give each of the 5 ponds one fish to ensure the "at least one" requirement. This uses 5 fish, leaving 10 fish to distribute.
Step 2: Now count the nonnegative integer solutions to y1 + y2 + y3 + y4 + y5 = 10 (where yi is the additional fish in pond i). The number of solutions is C(10 + 5 - 1, 5 - 1) = C(14,4).
Step 3: Compute the combination: C(14,4) = (14 × 13 × 12 × 11) / (4 × 3 × 2 × 1) = 1001.
Answer: 1001