Someone please help me answwer these math questions and expain how you got the answer..since im confused?

Question

Here are the problems I'm not sure how to solve...Probability Problems:

How many arrangements are there for the letters in the word MULTIPLY?

A club contains 7 boys and 7 girls. In how many ways can a group of four students be seleected if 3 boys and 2 girls are in it?

From a regular 52 deck of cards and a number cube what is the probability that....
aAred card is drawn and a number at least 3 is rolled?
A spade is drawn then a 1 is rolled?

A bag contains 5 red, 9 blue, and 6 green marbles.
P(green, then green, then green) given the marble is replaced after it is drawn.
P(the same color each of three selections) given the marble selected is replaced.

How many different license plates can be made is each plate contains four letters from A to J then three digits?

A committee contains 7 Democrats and 7 Republicans. find the probability a sybcommittee of 5 people will contain 2 DEmocrats and 3 Republicans.

please explain how you find the aanswer to these problems

Answer 1

1. "MULTIPLY"
No of arrangement = 8!/2!= 20160
For different n objects, the no of arrangement is given to be n!
Within the n objects, supposed there are k objects that are same, then the no of arrangement is n!/k!

2. Something wrong with your quetion, you want a group of 4 student with 3 boys and 2 girls????

3. If 2 events are independent, the just multiply the probability.
Of course, whatever you do to the card wont affect whatever you do to the cube.
Thus P(red card drawn and at least 3 rolled) = P(red card drawn) X P(at least 3 rolled) = 1/2 * 2/3 = 1/3
P(spade drawn andd 1 rolled) = P(spade drawn) X P(1 rolled) = 1/4 * 1/6 = 1/24

4. Since the marbles are replaced after it is drawn, the probability of drawing a random colour after one is drawn is independant. So just multiply them with the initial probabilioty.
P(G G G) = (6/20)^3 = 0.027
P(3 same colour) = (6/20)^3 + (9/20)^3 + (5/20)^3 = 0.13375




5. so you have ****^^^ where * is an alphabet, ^ is number. There are10 choices for *. For ^ take not that 0 is not a valid number on a license plate(that doesnt occur in most countries). So u have 1-999 ie 999 numbers.
So for alphabet, you have 10*10*10*10 choices. For number u have 999 choices. Just multiply them.
Thus no of choices is : 10^4 * 999

6. A group of 5 choosing people. Choosing 2 democrats from 7 democrats, the number of choices is 7C2
Choosing 3 rep from 7 rep, the no of choices is 7C3.
(Binomial theorem)
Thus, the no. of choices 2 make a subcom of 5 ppl with 2 dem and 3 rep is 7C2 * 7C3
The no of choices making an arbitrary subcom of 5 ppl from 14 ppl is 14C5
Thus Probability = 7C2*7C3 / (14C5)

Answer 2

8!/2!

A club contains 7 boys and 7 girls. In how many ways can a group of five students be selected if 3 boys and 2 girls are in it?
C(7;3)*C(7;2)
=735