Example 45

EXAMPLE 45 Number of permutations of nondistinct items

Brandon brings a healthy snack to school each day, consisting of 5 carrot sticks, 4 celery sticks, and 2 cherry tomatoes. If Brandon eats one item at a time, in how many different ways can he eat his snack?

Solution

We are seeking the number of permutations of $n = 11$ items, of which $n_{1} = 5$ are carrot sticks, $n_{2} = 4$ are celery sticks, and $n_{3} = 2$ are cherry tomatoes. Using the formula for the number of permutations of nondistinct items,

$\frac{n!}{n_{1}! \cdot n_{2}! \cdot n_{3}!} = \frac{11!}{5! \cdot 4! \cdot 2!} = \frac{39, 916, 800}{120 \cdot 24 \cdot 2} = 6930$

There are 6930 distinct ways in which Brandon can eat his snack.

NOW YOU CAN DO

Exercises 41 and 42.