**The Generalized Basic Principle of Counting**

**The Generalized Basic Principle of Counting**

If *r* experiments that are to be performed are such that the first one may
result in any of *n*_{1} possible outcomes, and if for each of these *n*_{1}possible outcomes there are *n*_{2} possible outcomes of the second experiment,
and if for each of the possible outcomes of the first two experiments there
are *n*_{3} possible outcomes of the third experiment, and if, ..., then there
is a total of
possible outcomes of the
*r* experiments.

**Example**- A college planning committee consists of 3 freshmen, 4 sophomores, 5 juniors,
and 2 seniors. A subcommittee of 4, consisting of 1 person from each class, is
to be chosen. How many different subcommittees are possible?

*Solution:*- We may regard the choice of a subcommittee as the combined outcome of the four spearate experiments of choosing a single representative from each of the classes. Hence it follows from the generalized version of the basic principle that there are possible subcommittees.

**Example**- How many different 7-place license plates are possible if the first 3 places are
to be occupied by letters and the final 4 by numbers?

*Solution:*- By the generalized version of the basic principle the answer is .