From the set $/{1, 2, 3, / dots, 20/},$ ten numbers are chosen at random, forming a subset. Let $M$ be the largest element among the ten numbers. Find the expected value of $M.$

The largest element can be as small as 10, which happens when the subset is {1, 2, ..., 10}. The probability of choosing this subset is (1/2) ^10 = 1/1024. (Every element from 1 to 10 can either be in the subset, or not.)

The largest element can also be 11. All the numbers in the subset must be from 1 to 10, and we must choose 1 to leave out, so the probability that the largest element is 11 is C (10,1) * 1/1024.

The largest element can also be 12. All the numbers in the subset must be from 1 to 11, and we must choose 2 to leave out, so the probability that the largest element is 12 is C (11,2) * 1/1024.

We can do the other cases similarly:

Largest element is 13 - > C (12,3) * 1/1024

Largest element is 14 - > C (13,4) * 1/1024

Largest element is 15 - > C (14,5) * 1/1024

Largest element is 16 - > C (15,6) * 1/1024

Largest element is 17 - > C (16,7) * 1/1024

Largest element is 18 - > C (17,8) * 1/1024

Largest element is 19 - > C (18,9) * 1/1024

Largest element is 20 - > C (19,10) * 1/1024

Adding these up, we get (1 + C (10,1) + C (11,2) + ... + C (19,10)) * 1/1024. Since 1 = C (9,0), we also get (C (9,0) + C (10,1) + C (11,2) + ... + C (19,10)) * 1/1024.

By the Hockey Stick Identity, C (9,0) + C (10,1) + C (11,2) + ... + C (19,10) = C (20,10), so the expected value of the largest element is 1/11*C (20,10) * 1/1024 = 4199/256.