A store clerk is stocking a shelf with boxes of cereal Suppo

A store clerk is stocking a shelf with boxes of cereal. Suppose that the clerk has 3 boxes of Fruit Loops, 5 boxes of Trix, and 4 boxes of Fruity Pebbles. Boxes of the same brand of cereal look the same.

a) How many distinct ways can the clerk arrange the cereal boxes on the shelf?

b) What is the probability that a box of Fruity Pebbles is at the beginning of the shelf and at the end?

c) How many ways can he stock the shelf such that the boxes of Trix are all together?

d) What is the probability that all of the boxes of Trix are together?

e) What is the probability that all of the boxes of each type of cereal are together? f) What is the probability that no boxes of Fruit Loops are next to each other?

Solution

a)

By permutation of like objects, there are

#ways = 12!/[3!5!4!] = 27720 ways [answer]

************

b)

There are 4C2 = 6 ways choose 2 boxes of Fruity to put at the ends.

Meanwhile, there are now just 10!/[3!5!2!] = 2520 ways to permute the other 10.

Thus, there are a total of 2520*6 = 15120 ways to do that.

Hence,

P = 15120/27720 = 0.545454545 [ANSWER]

***********

c)

We treat the Trix boxes as just 1 box.

Thus, we permute just 8 \"boxes\" now.

There are 8!/[1!4!3!] = 280 ways. [answer]

**************

d)

P(tric together) = 280/27720 = 0.01010101 [answer]

******************

Hi! Please submit the next part as a separate question. That way we can continue helping you! Please indicate which parts are not yet solved when you submit. Thanks!