Math test

MAT 1160 – Mathematics, A Human Endeavor

FALL SEMESTER 2020

Instructor: Dr. Gregory Galperin

Name (Last, First - printed):

I did my work by my myself and no one helped me.

Signature: Date:

T E S T # 3 : COMBINATORICS

Show all work. Do not forget to circle your answers.

1. How many positive (a) 10-digit; (b) 11-digit integers with all distinct digits are there?

Answer: (a) Answer: (b)

2. You roll 2 dice and then toss 3 coins; then you write the outcome of the five signs obtained (like 16HHT or 52THT). How many different outcomes can you get?

Answer:

3. When you call in Charleston, Champaign, or Chicago, you first dial the area code 235, or 581, or 549, or 508, or 312, or 773, followed by some 4 digits. How many telephone numbers can you dial this way, if the last 4 digits

(a) can be arbitrary; (b) cannot be 0?

Answer: (a) Answer: (b)

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4. The 5 cities are located at the vertices of a pentagon ABCDE. 2 roads connect A and B, 3 roads connect B and C, 4 roads connect C and D, 5 roads connect D and E, and 6 roads connect E and A. You want to get to the city C from the city A. In how many different ways can you do this task? Draw the pentagon ABCDE and justify your answer.

Answer:

5. (a) In how many ways can 4 desks be chosen from 12 desks? Answer:

(b) In how many ways can 4 students sit at 4 desks out of 12 desks? Answer:

6. The deck of 52 cards consists of 26 RED cards and 26 BLACK cards.

(a) In how many ways can one choose 22 RED cards from the deck? Answer:

(b). In how many ways can one choose from the deck 4 RED cards followed by 5 BLACK cards? Answer:

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7. In how many ways can 6 green marbles, 4 red marbles, 3 yellow marbles, and 5 white marbles be line up? All the 18 marbles have the same size and can be line up arbitrarily (chaotically).

Answer:

8. Some row of Pascal’s triangle looks as follows (where the figures stand for the respec- tive numbers in that row):

1, �, F , � , ... , 201 , 1 .

Find the numbers �, F , �, and the maximal number MAX in this row.

Justify your answer!

Hint: Remember the meaning of the number situated next to the very left 1. Also: the maximal number MAX is located on the vertical axis of symmetry of the Pascal’s triangle, which is the middle of the row in question.

Answer: � = ; F = ; � = ; MAX = .

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9. The locker’s secret combination is an unknown 7-digit integer each digit of which takes any of the four first prime numbers: 2, 3, 5, or 7. How many different combinations should you try to get the only one true combination that unlocks this locker?

Answer:

10. A license plate consists of 3 letters and 4 digits. How many such license plates do there exist, if

(a) the 4 digits followed by the 3 letters? Answer:

(b) the 4 digits and the 3 letters are located randomly? Answer: