Part 1.] identify the following series as arithmetic, geometric, both, or neither. includes picture
part 2.] evaluate c(3, 3)
a.] 1
b.] 6
c.] 24
part 3.] evaluate c(6,3)
a.] 20
b.] 120
c.] 720
part 4.] how many subsets of four elements each exist in a set of seven elements?
a.] 4!
b.] c(7, 4)
c.] p(7, 4)
part 5.] what is the probability of having 3 children that are all boys?
a.]
b.]
c.]

These are five questions and five answers.

Part 1.] Identify the following series as arithmetic, geometric, both, or neither. Includes picture

arithmetic

Justification:

The arithmetic series are those in which the distance (difference) between any consecutive terms is constant, so you can find the term An adding the difference to the previous term, A(n-1).

In this case the series is:

∞
∑ (3 + na) =>
n=1

Term1: 3 + a
Term2: 3 + 2a
Term3: 3 + 3a
Term3: 3 + 4a

As you see, the difference between two consecutive terms is a, which shows that it is an arithmetic series.

Part 2.] Evaluate C(3, 3)
A.] 1
B.] 6
C.] 24

option A) 1.

Explanation:

The formula for combinations is Cm,n = m! / [n! (m - n)! ]

So, for C3,3, C3,3 = 3! / [3! 0!] = 3! / 3! = 1.

From this you might learn that Cm,m for any m is always 1.

Part 3.] Evaluate C(6,3)
A.] 20
B.] 120
C.] 720

option A) 20

Explanation:

Use the same formula, Cm,n = m! / [n! (m - n)! ]

C6,3 = 6! / [3! 3! ] = 6*5*4 / (3*2*1) = 5*4 = 20

Part 4.] How many subsets of four elements each exist in a set of seven elements?
A.] 4!
B.] C(7, 4)
C.] P(7, 4) 

option B) C(7,4)

Explanation

You have to select collections of 4 elements from a set of 7 elements, and must realize that the order is irrelevant, i.e. it is the same the set ABCD as the set ACBD or DCAB or any combination of those four elements. So, the answer is a combination and not a permutation.

The number of subsets of 4 elements that you can form with 7 elements is: C(7,4), which is the option B.

Part 5.] What is the probability of having 3 children that are all boys?
A.] 1/2
B.] 1/8
C.] 3/8

option B: 1/8

Explanation:

The number of possible outcomes is 2 * 2 *  2 = 8

Only one of those outcomes is for all boys = 1

Therefore the probability is 1/8

You can see that representing the possible outcomes using B for boys and G for girls:

1) BBB (three boys)
2) BBG

3) BGB
4) BGG

5) GBB
6) GBG

7) GGB
8) GGG

Those are the 8 possible outcomes and as seen only one is for 3 boys.