A certain kind of combination-style lock shows 15 different numbers on the dial.

Question

A certain kind of combination-style lock shows 15 different numbers on the dial. To open the lock, the user must know the three-number sequence that is the "combination" for the lock. For example, to open the lock one might have to dial the three numbers 12-2-14 in the order shown here (first dialing 12, then 2, then 14). On this sort of lock it is possible use the same number twice, but you can't have two consecutive numbers that are the same. Example: You can have 7-10-7 but you can't have 7-7-10.

(a) How many different lock combinations are possible?


(b) Jim remembers that his combination includes the numbers 5, 9, and 13, but he can't remember what order. How many different combinations would he have to try to cover all the possibilities?


(c) Alice remembers that her combination includes the numbers 12 and 14 and some other number (different from 12 and 14). How many different combinations would she have to try to cover all the possibilities?


(d) Martha remembers that her combination includes exactly two 5's. How many different combinations would she have to try to cover all the possibilities?


(e) How many different combinations are there in which the combination consists of three different numbers and in which the numbers occur in order of increasing magnitude? For example, 5-10-13 but not 10-5-13. (Hint: You will *not* use permutations to find the answer.)

Explanation / Answer

a) there are 15 possibilities for the middle number

and 1st and 3rd number have 14 possibiliies

so total combinations possoble are

14* 15 * 14 = 2940

b) there are 3 numbers to try from

so first number can be one of 3

and then 2 and then 1

total possobilites are 3* 2 = 6

c) alice has 12 and 14 and she does not remember the 3rd number

so she can try in 6 differen order

but 3rd number can be chosen out of left 13 numbers

so total possibilites are 6* 13 = 26

d) if she has 2 5's then it must be at 1st and 3rd place

so she has 14 combinations to try

e) diffrent combinations if combination consists of 3 different numbers are

15 C 3

455 combinations

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