Question

Exercise 1.2. You have a pile of balls consisting of 4 red balls, 5 blue balls, 3 green balls, and 8 orange balls. Suppose you randomly choose a ball, one at a time, from this pile until you have pulled out a collection 4 balls. Assume repeats are allowed in all parts of this problem.
1. In how many ways could you make a selection that has exactly one green ball?
2. In how many ways could you make a selection that has exactly three green balls?
3. In how many ways could you make a selection that has no green balls?
4. In how many ways could you make a selection that has four green balls? (This is not a typo.)
Exercise 1.3. How many PIN numbers can we make if repeats are allowed? If repeats are not allowed?
Exercise 1.4. Answer each of the following.
1. How many 5-letter codes are there without repeats? (Recall there are 26 letters in the alphabet that we could use.)
2. How many of these codes start with the letter “a”?
Exercise 1.5. How many 3-letter codes can be made using only vowels? What about only consonants?
Exercise 1.6. There are 10 red jelly beans, 8 blue jelly beans and 12 green jelly beans in a jar.
1. How many ways could I pick out any combination of three jelly beans?
2. If I pick out three jelly beans, in how many ways could I pick exactly two red or exactly two green jelly beans?
3. If I pick out three jelly beans, in how many ways could I pick exactly one blue jelly bean?
Exercise 1.7. In how many ways could you draw a red card or a face card from a standard deck of 52 cards? What about a red card then a face card?

Answer 1

Note : Only one question is alllowed to be solved per post. Question 1.4 and 1.5 solved

Exercise 1.4. Answer each of the following.
1. How many 5-letter codes are there without repeats? (Recall there are 26 letters in the alphabet that we could use.)

The first letter of the codes can be selected in 26 ways.
The second letter of the codes can be selected in 25 ways.(Since one alphabet is used in the first case)
The third letter of the codes can be selected in 24 ways.
The fourth letter of the codes can be selected in 23 ways.
The fifth letter of the codes can select in 22 ways.

Hence the total number of ways =

2. How many of these codes start with the letter “a”?
The first letter of the codes can be selected in 1 way as it has to be 'a'.
The second letter of the codes can be selected in 25 ways.(Since one alphabet is used in the first case)
The third letter of the codes can be selected in 24 ways.
The fourth letter of the codes can be selected in 23 ways.
The fifth letter of the codes can be selected in 22 ways.

Hence the total number of ways =

Exercise 1.5. How many 3-letter codes can be made using only vowels? What about only consonants?

The vowels are a,e,i,o,u

Assuming repeats are allowed

3 letter codes made by vowels = 5^3 = 125

3 letter codes made by consonants = 21^3=9261