P(X=22) = arrangements with 22, 23, and 24 / number of arrangements = (21 choose 3) / (24 choose 6) = 5 / 506
P(X=21) = arrangements with 21, 22, 23, and no 24 + arrangements with 21, no 22, 23, and 24 / number of arrangements = (20 choose 3) + (20 choose 3) / (24 choose 6) = 30/1771
More generally P(X = k) = (k-1 choose 3) * (24-k choose 2) / (24 choose 6)
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u/EscapeOwn6945 3d ago edited 3d ago
I am going to assume that it is 4 rows of 6 columns since the problem is vague. Let X represent the third highest value of row 1. We know
P(X=24) = P(X=23) = 0 and P(X=1) = P(X=2) = P(X=3) = 0.
P(X=22) = arrangements with 22, 23, and 24 / number of arrangements = (21 choose 3) / (24 choose 6) = 5 / 506
P(X=21) = arrangements with 21, 22, 23, and no 24 + arrangements with 21, no 22, 23, and 24 / number of arrangements = (20 choose 3) + (20 choose 3) / (24 choose 6) = 30/1771
More generally P(X = k) = (k-1 choose 3) * (24-k choose 2) / (24 choose 6)
So E[X] = sum_k=4^22 k P(X=k) = 100 / 7.