# Thread: How many possible collection of toys?

1. ## How many possible collection of toys?

John will be given 4 chances to randomly pick toys from a bag containing 2 Ironmans, 3 Supermans, and 2 Batmans.

How many possible collection of toys can he pull out?

HELP please ... I tried to solve this many times but can't get the correct answer. BTW, 35 is the answer.

2. ## Re: How many possible collection of toys?

Hello, Kaloda!

John will be given 4 chances to randomly pick toys from a bag
containing: 2 Ironmans, 3 Supermans, and 2 Batmans.

How many possible collections of toys can he pull out?

Answer: 35.

There is no formula for this problem.
We must construct an exhaustive list.

$\text{Let }(x,y,z)\text{ represent: }x\text{ Ironmans, }y\text{ Supermans, }z\text{ Batmans.}$

$\begin{array}{ccccccc}(2,1,1): & {2\choose2}{3\choose1}{2\choose1} &=& 1\cdot3\cdot2 &=& 6 \\ \\[-4mm] (1,2,1): & {2\choose1}{3\choose2}{2\choose1} &=& 2\cdot3\cdot2 &=& 12 \\ \\[-4mm] (1,1,2): & {2\choose1}{3\choose1}{2\choose2} &=& 2\cdot3\cdot1 &=& 6 \\ \\ (2,2,0): & {2\choose2}{3\choose2}{2\choose0} &=& 1\cdot3\cdot1 &=& 3 \\ \\[-4mm] (2,0,2): & {2\choose2}{3\choose0}{2\choose2} &=& 1\cdot1\cdot1 &=& 1 \\ \\[-4mm] (0,2,2): & {2\choose0}{3\choose2}{2\choose2} &=& 1\cdot3\cdot1 &=& 3 \\ \\ (1,3,0): & {2\choose1}{3\choose3}{2\choose0} &=& 2\cdot1\cdot1 &=& 2 \\ \\[-4mm] (0,3,1): & {2\choose0}{3\choose3}{2\choose1} &=& 1\cdot1\cdot2 &=& 2 \\ \hline &&& \text{Total:} && 35 \end{array}$

3. ## Re: How many possible collection of toys?

It is not that hard if you do the math.