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