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# Probability, counting and combination

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1) When dealing with probability using a dice, let's assume we have a two fair, 6-sided dice that are rolled together.

a) What is the probability of the sum of the numbers on the dices to be 8 or larger?

b) What is the probability of rolling a 12 on the dices?

c) If one dice is red, and the other is green, what is the probability that the number on the red dice is greater than or equal to the one on the green dice?

d) Now assuming we have one more blue dice, all of different color (i.e. red, green, blue).What is the probability the number on the red dice is greater than or equal to the green dice and the blue dice?

e) What is the probability of all three dices having the same number?

2) (a) Is 0!=1! a correct statement? What is the answer for 0?

(b) Let us consider another example of sticks and stones. You now have 40 sets of sticks and stones. You want to take 20 sets with you while going for a small building project. In how many ways can you do it?

(c) Find the number of subsets of the set {1,2,3,4,5,6,7,8,9,10,11,12} having 5 elements.

(d) We have 50 points that are painted on a circle. How many cyclic quadrilaterals can be drawn by using these different points? (Hint: For any set of 4 points, we get a cyclic quadrilateral.)

(e) There are 500 novels and 40 biographies. In how many ways can 4 novels and 2 biographies be arranged on a shelf?

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1) When dealing with probability using a dice, let's assume we have a two fair, 6-sided dice that are rolled together.

a) What is the probability of the sum of the numbers on the dices to be 8 or larger?
First, there are 6*6=36 possible outcomes if we roll two fair dice. The outcomes with sum of 8 or larger is (2,6), (3,5), (3,6), (4,4), (4,5), (4,6), (5,3), (5,4), (5,5), (5,6), (6,2), (6,3), (6,4), (6,5) and (6,6). So there are 15 such outcomes. So probability=15/36.

b) What is the probability of rolling a 12 on the dices?
First, there are 6*6=36 possible outcomes if we roll two fair dice. The outcomes with sum of 12 is (6,6). So there are 1 such outcomes. So probability=1/36.

c) If one dice is red, and the other is green, what is the probability that the number on the red dice is greater than or equal to the one on the green dice?
First, there are 6*6=36 possible outcomes if we roll two fair dice. The outcomes when number in red dice is greater than one in green dice is (2,1), (3,1), (3,2), (4,1), (4,2), (4,3), (5,1), (5,2), (5,3), (5,4), (6,1), (6,2), (6,3), (6,4) and (6,5). So there are 15 such outcomes. So probability=15/36.

d) Now assuming we have one more blue dice, all of different color (i.e. red, green, blue).What is the probability the number on the red dice is greater than or equal to the green dice and the blue dice?
First, there are 6*6*6=216 possible outcomes if we roll three fair dice. The outcomes when number in red dice is greater than ones in other two dices will be this: when we get 1 in red dice, there is one possible outcome; when we get 2 in red dice, there is 22=4 possible outcome; when we get 3 in red dice, there is 32=9
possible outcome; when we get 4 in red dice, there is 42=16 possible outcome; when we get 5 in red dice, there is 52=25 possible outcome; when we get 6 in red dice, there is 62=25 possible outcome. Hence, there are totally 1+4+9+16+25+36=91 such outcomes. Hence, probability=91/216.

e) What is the probability of all three dices having the same number?
First, there are 6*6*6=216 possible outcomes if we roll three fair dice. The outcomes when all three dices have the same number is (1,1,1), (2,2,2), (3,3,3), (4,4,4), (5,5,5) and (6,6,6). So there are 6 such outcomes. So probability=6/216=1/36.

2)

(a) Is 0!=1! a correct statement ? What is the answer for 0! ?
No, it is not a correct statement. The definition is 0!=1 but you can not write 0!=1!.

(b)Let us consider another example of sticks and stones. You now have 40 sets of sticks and stones. You want to take 20 sets with you while going for a small building project. In how many ways can you do it?
You have C(40,20) ways to do it where C is symbol of combination.

(c) Find the number of subsets of the set {1,2,3,4,5,6,7,8,9,10,11,12} having 5 elements.
The number of subsets is C(12,5)=792 where C is symbol of combination.

(d)We have 50 points that are painted on a circle. How many cyclic quadrilaterals can be drawn by using these different points? (Hint: For any set of 4 points we get a cyclic quadrilateral.)
There are C(50,4)=230300 cyclic quadrilaterals where C is symbol of combination.

(e) There are 500 novels and 40 biographies. In how many ways can 4 novels and 2 biographies be arranged on a shelf?
First, we choose 4 from 500 novels and 2 from 40 biographies. There are C(500,4) and C(40,2) where C is symbol of combination. After choosing 6 books, there are 6! Ways to arrange them on a shelf where ! Is symbol of factorial. So totally, there are C(500,4)*C(40,2)*6! Ways.

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