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.

the first successful, urban blues recording, selling more than 75,000 copies, was made in year 1920.

mamie smith recorded the first commercially successful example of the urban blues,   with the recording of “crazy blues”."crazy blues" was recorded on august 10, 1920, in new york city. within a month of the release of “crazy blues” it had sold 75,000 copies.

answer; institution based;