On a probability problem from 投资理财

Problem: A standard cubical die is thrown four times, with respective outcome a,b,c,d. what is the probability that  a≤b≤c≤d ? http://bbs.wenxuecity.com/tzlc/516710.html

Solution: We label six urns as 1, 2, 3, 4, 5, and 6 and place four indistinguishable balls in the urns. Each placement of the balls in the urns represents a successful outcome a≤b≤c≤d.For instance, suppose we have one ball in the 2nd urn, two balls in the 5th urn, and one ball in the 6th urn. This arrangement corresponds to the outcome a=2, b=5, c=5, and d=6.

There are C(4+6-1, 6-1)= C(9,5) = 126 ways to place 4 idential balls in 6 distinct urns. Hence,

The probability = 126/(6^4) = 7/72.

Comment: It probably takes a top MATHCOUNTS mathlete one or two minutes to solve the problem.