If n \gt 7 then what is the probability that C(n,7) is a multiple of 7?
- A. 0
- B. \frac{1}{7} ✓
- C. \frac{1}{2}
- D. 1
Correct Answer: B. \frac{1}{7}
Explanation
By Lucas' Theorem, \binom{n}{7} is a multiple of 7 if the base-7 digit of n corresponding to 7^1 is less than the corresponding digit of 7 (10_7). Thus, the 7^1 digit of n must be 0. Since n can have any remainder modulo 49, this specific digit is 0 for exactly 1 out of every 7 consecutive intervals. The probability is \frac{1}{7}.
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