To count the number of pages in between these two page numbers, we want to subtract  and then add . We have to add  because if we don't, we end up not counting either the first or last page of the selection. 

Therefore, our answer is 

An important consideration whenever you're asked for the number of integers between two other numbers is to determine whether the two "end points" are included. Here the word "inclusive" tells us that we need to count both -8 and 17 in our calculation.

A helpful tip for these questions: if you're including both end points, you can calculate by using the range (greatest value minus smallest value) and adding 1 to account for the inclusive set.  (If you need to prove that to yourself, just use "how many integers between 1 and 3 inclusive" - clearly the set is 1, 2, 3 for three integers, but 3-1 = 2, so you can see you need to add one more to get the answer).

For this question, that leaves us with 17 - (8) = 25 as the range, and if you add 1 to that for "inclusive" you get the correct answer, 26.

Another way to look at this one is to chop it up: between -8 and 17 there are 8 negative integers (-1 through -8), 17 positive integers (1 through 17), and 0 as one more integer.  8 + 17 + 1 = 26. 

A prime numbers is a number greater than 1 that can only be divided by 1 and itself. Simply count from 1 to 25 and see how many values fit the criteria.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

Prime numbers are underlined. Nine prime numbers are in this set interval.

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.)