A lock has five buttons 12345 A key for the lock is a sequen

A lock has five buttons 1,2,3,4,5. A key for the lock is a sequence of five numbers. How many keys are possible? What is the probability of guessing the key? What is the probability of guessing the key in three or fewer tries? How many keys are possible if no number is repeated? How many keys are possible if two buttons are pressed together, then the other three in sequence?

Solution

ai) The possible no. of keys = 5^5 = 3125;

ii)Probability of guessing a key = 1/3125 = 0.00032

iii)assuming guessing of key is dependent on the previous trials

Guessing the key in three or fewer times= guess correctly in first time + guresscorrectly in the second time/first is wrong guess + guess correctly in third time/first two times guess wrong

probability of guessing the key three or fewer times = (1/3125) +(3124/3125)*(1/3124) + (3124/3125)*(3123/3124)*(1/3123)=0.00096

b) If no number is repeated then the number of keys possible are 5! = 120

c) if two buttons are pressed together, then the number of ways to select those two buttons from 5 numbers = 5C2 = 10

the number of keys are possible if two buttons are pressed together then the other three in sequence is

= 10*3! = 60