The probability that a patient recovers from a delicate heart operation is 0.9 what is the probability that exactly 5 of the next 7 patients having the operation survive?

C(7,2)*(.9)^5*(.1)^2, or about .124 = 12.4%

For the desired outcome, considering the seven patients, you need:

(Survive,Survive,Survive,Survive,Survive,Die,Die)

(Survive,Survive,Survive,Survive,Die,Survive,Die)

(Survive,Survive,Survive,Die,Survive,Survive,Die)

.

.

.

(Die,Die,Survive,Survive,Survive,Survive,Survive)

There are C(7,2) [the number of combinations of 7 things taken 2 at a time] = 21 possible desired outcomes. The probability of each of these outcomes is (.9)*(.9)*(.9)*(.9)*(.9)*(.1)*(.1). Multiplying 21 by (.9)*(.9)*(.9)*(.9)*(.9)*(.1)*(.1) yields the answer.