Problem 1 (20 points) 1. [12 points] Consider the numeric keypad shown alongside

Question

Problem 1 (20 points) 1. [12 points] Consider the numeric keypad shown alongside that can be used to enter a PIN. AG 4-digit number such as 2356 may correspond to a variety of strings like CELO, BFKO, etc. a) How many 4-letter strings exist that correspond to the number 4567? What about 1114 when no letter can be repeated twice? b) How many 4-digit PINS can be created that contain no repeated digit? How may4 wr digit PINS start with 2 or end with 3? 2. [8 points] A student committee consisting of a head, a member and a secretary must be formed by choosing among Ahmed, Fatma, Aisha, and Salem. Suppose that Salem cannot be the head and either Aisha or Ahmed must be member. a) Draw a tree showing the different ways the committee can be formed. b) How does the tree look like if, in addition to the above, we are told that Fatma can only be either a Problem 2 (15 points) Consider a 4x4 grid containing the four colors like the one shown alongside. For each of the following cases, find out how many different such grids exist. In each case justify yourR BYR BRRG B GYY reasoning a) The diagonal contains no repeated colors b) All rows have no repeated colors. c) The first row contains one instance of the pattern RB. d) The colors in the corner positions are either G or Y e) There are exactly 3 R's and 5 G's Problem 3 (20 points) Suppose you throw four dice, one after the other. Answer the following: a) b) c) d) e) How many ways are there so that all dice show a different number? How many ways are there so that two of the dice are the same and the other two are different? How many ways are there so that all dice show an even number? How many ways are there so that the first two dice are equal to 7 and the last two are different? How many ways are there so that half of the dice are even and the other are odd?

Explanation / Answer

Solution:

Note:
The first question is done as per Chegg guidelines, please repost others.

1)

a)

Number of combinations possible will be

for 4 letters string

3^4 = 81 string combinations are possible.

1114 cannot have unrepeated letters since there is only q and z on the button which means if we press it three times any of q or z will be repeated.

b)

Number of PINS possible with no repetition

10*9*8*7 = 5040

consider it like a 4 boxes where we can place the digits so in the first box we can put 10 digits and in the second 9 since the one which is placed in the first won't be repeated here and so on

start with 2 or end with 3

here it is not mentioned if the repetition is allowed, we are considering it ins not allowed

start with 2 can happen in 9*8*7 = 504 ways

end with 3 can also happen in 504 ways

total number of ways= 1008

in case of repetition 2*9^3

I hope this helps if you find any problem. Please comment below. Don't forget to give a thumbs up if you liked it. :)

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