Question 3:

A snack cupboard contains seven chocolate bars (milk, 70%, 92%, hazelnut, vanilla, caramel, hot chilli), six different jars of peanuts, three different bags of chips, and four bags of pretzel sticks (original, gluten-free, pumpernickel, unsalted). In how many ways can a tray of snacks be put together if the tray must contain only one chocolate bar, one jar of peanuts, one bag of chips, and either

a) any number of bags of pretzel sticks, or (any possible combination of the 4 types of pretzel sticks)

b) one bag of pretzel sticks?

Provide an appropriate diagram with your solution process.

My answers are below:

I've already tried answering this question using multiplication (7x6x3x4 = 504). These are not acceptable answers according to the instructor.

*** Instructor comment: Hmm, the pretzel bags in Q3a... try writing out the combinations. I think you might get a few more! Tree diagrams are all about listing possibilities so nothing wrong with doing that (and it's a great way to explain something without any kind of formula).

Q3A:

First Choice	Second Choice	Third Choice	Fourth Choice
Chocolate Bar 1	Jar of Peanuts 1	Bags of Chips 1	Bag of Pretzels 4: ABCD
Chocolate Bar 2	Jar of Peanuts 2	Bags of Chips 2
Chocolate Bar 3	Jar of Peanuts 3	Bags of Chips 3
Chocolate Bar 4	Jar of Peanuts 4
Chocolate Bar 5	Jar of Peanuts 5
Chocolate Bar 6	Jar of Peanuts 6
Chocolate Bar 7
7 x	6 x	3 x	1 = 126

If the tray must contain only one chocolate bar, one jar of peanuts, one bag of chips, and (any number of my choosing) 4 bags of pretzel sticks, the tray can be put together in 126 different ways.

First Choice	Second Choice	Third Choice	Fourth Choice
Chocolate Bar 1	Jar of Peanuts 1	Bags of Chips 1	Bag of Pretzels 2: AB
Chocolate Bar 2	Jar of Peanuts 2	Bags of Chips 2	Bag of Pretzels 2: AC
Chocolate Bar 3	Jar of Peanuts 3	Bags of Chips 3	Bag of Pretzels 2: AD
Chocolate Bar 4	Jar of Peanuts 4		Bag of Pretzels 2: BC
Chocolate Bar 5	Jar of Peanuts 5		Bag of Pretzels 2: BD
Chocolate Bar 6	Jar of Peanuts 6		Bag of Pretzels 2: CD
Chocolate Bar 7
7 x	6 x	3 x	6 = 756

If the tray must contain only one chocolate bar, one jar of peanuts, one bag of chips, and (any number of my choosing) 2 bags of pretzel sticks, the tray can be put together in 765 different ways.

First Choice	Second Choice	Third Choice	Fourth Choice
Chocolate Bar 1	Jar of Peanuts 1	Bags of Chips 1	Bag of Pretzels 3: ABC
Chocolate Bar 2	Jar of Peanuts 2	Bags of Chips 2	Bag of Pretzels 3: ACD
Chocolate Bar 3	Jar of Peanuts 3	Bags of Chips 3	Bag of Pretzels 3: BCD
Chocolate Bar 4	Jar of Peanuts 4
Chocolate Bar 5	Jar of Peanuts 5
Chocolate Bar 6	Jar of Peanuts 6
Chocolate Bar 7
7 x	6 x	3 x	3 = 376

If the tray must contain only one chocolate bar, one jar of peanuts, one bag of chips, and (any number of my choosing) 3 bags of pretzel sticks, the tray can be put together in 376 different ways.

Q3B:

First Choice	Second Choice	Third Choice	Fourth Choice
Chocolate Bar 1	Jar of Peanuts 1	Bags of Chips 1	Bag of Pretzels 1: A
Chocolate Bar 2	Jar of Peanuts 2	Bags of Chips 2	Bag of Pretzels 2: B
Chocolate Bar 3	Jar of Peanuts 3	Bags of Chips 3	Bag of Pretzels 3: C
Chocolate Bar 4	Jar of Peanuts 4		Bag of Pretzels 4: D
Chocolate Bar 5	Jar of Peanuts 5
Chocolate Bar 6	Jar of Peanuts 6
Chocolate Bar 7
7 x	6 x	3 x	4 = 504

If the tray must contain only one chocolate bar, one jar of peanuts, one bag of chips, and 1 bag of pretzel sticks, the tray can be put together in 504 different ways.