Question

A company representative is distributing 5 identical samples of a product among 12 houses in a row such that each house gets at most one sample. The probability that no two consecutive houses get one sample is:

• A. \(\frac{7}{99}\)
• B. \(\frac{5}{12}\)
• C. \(\frac{4}{13}\)
• D. \(\frac{5}{31}\)

Hint:

Use combinatorial formulas for counting selections with no two consecutive elements and calculate probabilities accordingly.

Answer 1

The Correct Option is A

Total ways to distribute 5 identical samples among 12 houses, with at most one sample each, is \(\binom{12}{5}\). Number of ways to select 5 houses with no two consecutive: Use the formula for non-consecutive selection: \(\binom{n - k + 1}{k}\), where \(n=12\), \(k=5\). \[ \binom{12 - 5 + 1}{5} = \binom{8}{5} = 56. \] Probability: \[ \frac{56}{\binom{12}{5}} = \frac{56}{792} = \frac{7}{99}. \]