The combination to a lock consists of a sequence of three numbers in the range 0
ID: 3268327 • Letter: T
Question
The combination to a lock consists of a sequence of three numbers in the range 0-38. a) How many combinations are possible in which no two consecutive numbers can be the same? b) How many combinations are possible in which all three numbers are different? c) What is the probability that the combination of a randomly-chosen lock consists of three different numbers? d) What is the probability that the combination of a randomly-chosen lock contains at least one repeated number? a) If no two consecutive numbers can be the same, write the expression (consisting of a product of numbers) that represents the number of possible combinations. 39 * 38 * 38 (Do not simplify.) Thus, if no two consecutive numbers can be the same, the number of combinations is 56318. (Simplify your answer.) b) If all three numbers are different, the number of combinations is 54834. (Simplify your answer.) c) The probability that the combination of a randomly-chosen lock consists of three different numbers is. (Round to two decimal places as needed.) d) The probability that the combination of a randomly-chosen lock contains a repeated number is. (Round to two decimal places as needed.)Explanation / Answer
a) Total number of combinations where no 2 consecutive numbers could be the same.
_ _ _
Now here we have 39 ways to fill the first blank as there are 39 digits from 0-38
Then we can fill the second blank in 38 ways because we cannot use the same digit as that in the first blank
And then we have 38 ways to fill the last blank as well. ( same logic )
Therefore total number of lock combinations possible are:
= 39*38*38 = 56316
Therefore 56316 is the required number of combinations possible.
b) Number of combinations where are 3 digits are different would be:
= 39*38*37 = 54834
Therefore 54834 is the required number of combinations possible.
c) Total number of combinations possible are: = 393 because there are 39 ways to fill each digit ( without any condition)
= 393 = 59319
Probability that a randomly chosen lock combination consists of three different digits
= Number of combinations with three different digits/ Total number of combinations possible
= 54834/59319 = 0.92
Therefore 0.92 is the required probability here.
d) Probability that a combination consist of a repeated number:
= 1- Probability that all three digits are different
= 1 - 0.92
= 0.08
Therefore 0.08 is the required probability here.