The Sum Rule
Suppose that A and B are two disjoint events, that is, they never occur together. Further, suppose that A occurs in m ways and B occurs in n ways. Then A or B can occur in m + n ways. This rule can be extended to more than two mutually exclusive events. Suppose you have 5 full-sleeves and 7 half-sleeves shirts. In how many ways can you wear a shirt? Since you have 5 + 7 = 12 shirts and you are going to wear exactly one shirt, you can wear a shirt in 5 + 7 = 12 ways. If in addition to 5 full-sleeves and 7 half-sleeves shirts, you have 8 T-shirts, then the number of ways in which you can wear a shirt out of these is 5 + 7 + 8 = 20 ways. The above illustration explains a general principle, called the sum rule.
Rachit is taken to a toy shop containing 8 distinct toy helicopters, 7 distinct aeroplanes and 20 distinct toy guns. If Rachit is allowed to choose exactly one of the toys in the shop, he will have to choose one toy out of 8 + 7 + 20 = 35 toys. That is, there are 8 + 7 + 20 =35 ways in which Rachit can choose a toy. The important things to remember is that if we have to choose exactly one object out of the several disjoint groups of object, then the number of ways of choosing that object is obtained by adding the number of objects in each of this distinct groups.