What's the probability that each shooter, A, B, C, respectively, will hit the target only once? Three shooters shoot at the same target. The probability for every shooter is $A, B, C$ respectively. What is the probability that the shooters will hit the target only once?
I know I can do something like:
$$P(X = 1) = P(A)P(B^c)P(C^c)+P(A^c)P(B)P(C^c)+P(A^c)P(B^c)P(C)$$
But is there any shorter way to solve this problems? .

It depends on what you consider shorter. You can do it with five instead of six multiplications using inclusion–exclusion):
$$P(X=1)=P(A)+P(B)+P(C)-2P(A)P(B)-2P(A)P(C)-2P(B)P(C)+3P(A)P(B)P(C)\;.$$
(Multiplying by an integer doesn’t require a multiplication since you can replace it by the corresponding number of additions.)
