There are 5 dogs and 3 cats. They all are arranged in a line. What is the…
2023
There are 5 dogs and 3 cats. They all are arranged in a line. What is the probability that there is one cat at both the ends of the line?
- A.
2/3
- B.
3/28
- C.
2/5
- D.
1/3
Attempted by 4 students.
Show answer & explanation
Correct answer: B
Final answer: 3/28
Step 1: Total arrangements of all 8 animals = 8!.
Step 2: Count favorable arrangements with a cat at each end: choose 2 cats out of 3 for the ends (C(3,2) = 3) and arrange them on the two ends in 2! = 2 ways, giving 3 × 2 = 6 choices for the ends.
Step 3: Arrange the remaining 6 animals in the middle in 6! ways.
Step 4: Favorable total = 6 × 6!. Probability = (6 × 6!)/8!.
Step 5: Simplify the fraction: (6 × 6!)/8! = 6/(8 × 7) = 6/56 = 3/28.
Alternate (sequential) method:
Pick the left end first: probability it is a cat = 3/8. Then the right end must be a cat: probability = 2/7. Multiply: (3/8) × (2/7) = 3/28.
Therefore the probability that there is one cat at both ends of the line is 3/28.