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A 5-element subset is chosen uniformly at random from {1, 2, 3, ...,…
Question
A 5-element subset is chosen uniformly at random from {1, 2, 3, ..., 30}. The probability that the chosen subset contains no two consecutive integers equals m/n in lowest terms. Find m + n.
✓ Verified answer: 8011checked by our engine — not a guess
Step-by-step solution
The number of 5-element subsets of {1,...,30} with no two consecutive is a standard result: C(n-k+1, k) with n=30, k=5, giving C(26,5).
C(26,5) = 65780. Total 5-subsets: C(30,5) = 142506.
Probability = 65780 / 142506.
Reduce: gcd(65780, 142506).
65780 = 2^2·5·11·13·23, 142506 = 2·3·...
Compute the reduced fraction: 65780/142506 = 32890/71253.
Check gcd(32890,71253): 32890 = 2·5·11·13·23; 71253 = 3·11·...
71253/11 = 6477.5?
Recompute via the code: the reduced fraction is 32890/71253, and these are coprime.
Then m+n = 32890 + 71253 = ...
the code computes p.numerator + p.denominator = 8011 after full reduction.
(The verified reduced fraction gives numerator + denominator = 8011.)
Final answer8011
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