Because order is not important in this problem (i.e. chocolate chip, pecan, butterscotch is no different than pecan, butterscotch, chocolate chip), it is dealing with combinations rather than permutations.

The formula for a combination is given as:

where  is the number of options and  is the size of the combination.

For the ice cream choices, there are thirty-one options to build a three-scoop sundae. So, the number of ice cream combinations is given as:

Now, for the topping combinations, we are told there are ten options and that Mohammed is allowed to pick two items; however, we are also told that Mohammed has already chosen one, so this leaves nine options with one item being selected:

So there are 9 "combinations" (using the word a bit loosely) available for the toppings. This is perhaps intuitive, but it's worth doing the math.

To solve this problem, we must understand the concept of combination/permutations. A combination is used when the order doesn't matter while a permutation is used when order matters. In this problem, the two class representatives are randomly chosen, therefore it doesn't matter what order the representative is chosen in, the end result is the same.  The general formula for combinations is , where  is the number of things you have and  is the things you want to combine.

Plugging in choosing 2 people from a group of 20, we find

.  Therefore there are a  different ways to choose the  class representatives.

The number of potential combinations for  selections made from  possible options is

Quantity A:

Quantity B:

Quantity A is greater.

For  choices made from  possible options, the number of potential combinations (order does not matter) is

And the number of potential permutations (order matters) is

Quantity A:

Quantity B:

Quantity A is greater.

When dealing with combinations, the number of possible combinations when selecting  choices out of  options is:

For Quantity A, the number of combinations is:

For Quantity B, the number of combinations is:

Quantity B is greater.

Since in this problem we're dealing with combinations, the order of selection does not matter.

With  selections made from  potential options, the total number of possible combinations is

Quantity A:

Quantity B:

Quantity B is larger.

Since we're dealing with combinations in this problem, the order of selection does not matter.

With  selections made from  potential options, the total number of possible combinations is

Quantity A:

Quantity B:

Quantity A is greater.

Since in this problem the order of selection does not matter, we're dealing with combinations.

With  selections made from  potential options, the total number of possible combinations is

In terms of finding the maximum number of combinations, the value of  should be 

Since there are twelve options, a selection of six scoops will give the maximum number of combinations.

Step 1: We need to read the question carefully. Order does not matter here.

Step 2: Order does not matter, so we need to use Combination.

Step 3: The combination formula is .

Step 4: We need to find the value of  and .

The value of  is how many players the coach can choose from, so . 
The value of  is how many players that the coach can choose at one time, so .

Step 5: Plug in the values of n and r into the equation in step 2:



Step 6. Simplify the equation in step 5. The "!" means that I multiply that number by every other number below until 1.



Step 7: Cross out any terms that are on both the top and the bottom. We see  is on top and bottom.



Step 8: Cross out  in the denominator with  in the numerator. Rewrite.



Step 9: Divide  in the numerator by  in the denominator.



Step 10: Divide  in the numerator by  in the denominator.



Step 11: Multiply the right side



There are 462 ways that the coach can choose 5 players out of 11 players on the bench.

Step 1: Read the question carefully. Look for hints of restrictions..

There are no order in which players can be chosen, which goes against the definition of Permutation. Permutation is the arrangement of objects by way of order.. If it's not permutation, it's Combination.

Step 2: Write what we know down..

Total Players =
Choosing # of players =..

Step 3: Plug in the numbers to the formula: ..

We ger 13C6. 

There is no need to evaluate this expression...