The fundamental counting principle says that if you want to determine the number of ways that two independent events can happen, multiply the number of ways each event can happen together. In this case, there are 5 * 7, or 35 unique combinations of pants & shirts Mark can wear. If he wears one combination each day, he can last 35 days, or 5 weeks, without buying new clothes.

This is a permutation problem, because we are looking for the number of groups of winners.  Consider the three positions, and how many choices there are for each position:  There are 20 choices for 1st place, 19 for 2nd place, and 18 for 3rd place.

20, 19, 18     

Multiply to get 6840.

Since this a combination problem and we want to know how many different ways the cookies can be created we can solve this using the Fundamental counting principle. 4 x 3 x 8 = 96

Multiplying each of the possible choices together.

The total number of possible combinations of a series of items is the product of the total possibility for each of the items.  Thus, for the letters, there are 26 possibilities for each of the 3 slots, and for the numbers, there are 10 possibilities for each of the 3 slots.  The total number of combinations is then: 26 x 26 x 26 x 10 x 10 x 10 = 17,576,000 ≈ 18 million.

The formula for the number of lines determined by n points, no three of which are “collinear” (on the same line), is n(n-1)/2. To find the number of lines determined by 8 points, we use 8 in the formula to find 8(8-1)/2=8(7)/2=56/2=28. (The formula is derived from two facts: the fact that each point forms a line with each other point, hence n(n-1), and the fact that this relationship is symmetric (i.e. if a forms a line with b, then b forms a line with a), hence dividing by 2.)

The first person holds 7 hands. The second holds six by virtue of already having help the first person’s hand. This continues until through all 8 people. 7+6+5+4+3+2+1=28.

There are 5x4x3 ways to arrange 5 flavors in 3 ways.  However, in this case, the order of the flavors does not matter (e.g., a cone with strawberry, mint, and banana is the same as a cone with mint, banana, and strawberry).  So we have to divide 5x4x3 by the number of ways we can arrange 3 different things which is 3x2x1.  So (5x4x3)/(3x2x1) is 10.  

One can also use the combination formula for this problem: _nC_r = n! / (n-r)! r!

Therefore: ₅C₃ = 5! / 3! 2!

= 10

(Note: an example of a counting problem in which order would matter is a lock or passcode situation.  The permutation 3-5-7 for a three number lock or passcode is a distinct outcome from 5-7-3, and thus both must be counted.)

You have 3 possible types of bread, 2 possible types of meat, and 2 possible types of cheese. Multiplying them out you get 3*2*2, giving you 12 possible combinations.

2 bread choices * 3 cheese choices * 5 meat choices = 30 sandwich choices

36 possible flavors * 3 possible sizes * 2 possible cones = 216 possible combinations.