Give a combinatorial proof that if n is a positive integer then ∑n k =0 k2 (n k ) = n(n + 1)2n−2.[Hint: Show that both sides count the ways to select a subset of a set of n elements together with two not necessarily distinct elements from this subset. Furthermore, express the righthand side as n(n − 1)2n−2 + n2n−1.]

Typically a point in a three dimensional cartesian coordinate system is represented by which of the following answers in the picture !

If each bricks costs and he can only buy one brick how much will it cost him to get the material to put around the outside of his garden?

Aking and his army will attempt to capture a fortress. the left and right flanks break off from the main group to attack the west and east guard towers. suppose the left flank has a 60% chance of success and the right flank has a 75% chance of success, independently of one another. if both flanks capture their respective targets, then the king has a 98% chance of successfully taking the fort. if, however, only the left flank captures its tower, the king has an 80% chance of success; if only the right flank succeeds, the king has a 50% chance. if both flanks fail, then the king's chance of capturing the fort drops to 20%. it turns out the king captures the fort. what is the probability that one, and only one, flank was successful (either the left, or the right, but not both)? (round your answer to 3 decimal places)