Questions 4-6

4.
Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and PR is parallel to the x-axis. The x and y coordinates of P, Q and R are to be integers that satisfy the inequalities -4 <= x <= 5 and 6 <= y <= 16. How many different triangles with these properties could be constructed?
 (A) 110
(B) 1,100
(C) 9,900
 (D) 10,000
 (E) 12,100

Solution:

No. of possible value for x=10 and y=11
 1) information given is right angle at P, PR(ll) to x axis
 information inferred y coordinates of p and r is same.
  2) PQ (llel) y therefore x coordinates of P and Q is same.
For P x can be selected in 10 ways and y in 11 ways = 10*11
 For R x in 9 ways and y in 1 way (as same of P) =9*1
For q x in 1 way and y in 10 ways (one already selected for P) =10*1
  Total ways=10*11*9*10=9900


5.
 A password of a computer used five digits where they are from 0 and 9. What is the probability that the password solely consists of prime numbers and zero?
A 1/32
B 1/16
C 1/8
D 2/5
E ½

Solution:
There are 10 possible options (0,1,2,3,4,5,6,7,8,9) for each digit.
5 of the options (0,2,3,5,7) are zero or prime.
 So, P(a given digit is zero or prime) = 5/10 = 1/2
A quick way is to look at this as an AND probability.
P(all five digits are zero or prime) = P(1st digit is zero or prime AND 2nd digit is zero or prime AND 3rd digit is zero or prime AND 4th digit is zero or prime AND 5th digit is zero or prime)
 This is equal to P(1st digit is zero or prime) x P(2nd digit is zero or prime) x P(3rd digit is zero or prime) x P(4th digit is zero or prime) x P(5th digit is zero or prime)
So, we get 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/32

6.

There are two set each with the number 1, 2, 3, 4, 5, 6. If randomly choose one number from each set, what is the probability that the product of the 2 numbers is divisible by 4?
Solution:
Picking 2 numbers from each set : 6c1*6c1=36  Favorable outcomes = 15
(1,4)
(2,2),(2,4),(2,6)
 (3,4)
 (4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
 (5,4)  (6,2),(6,4),(6,6)
Therefore Required probability = 15/36.