Chapter 13.2, Problem 63AYU

Combination Locks A combination lock displays 50 numbers. To open it, you turn clockwise to the first number of the “combination,” then rotate counterclockwise to the second number, and then rotate clockwise to the third number.

(a) How many different lock combinations are there?

(b) Comment on the description of such a lock as a combination lock.

To determine

To find:

a. How many different lock combinations are there?

Answer to Problem 63AYU

a.125000, if the numbers are repeated and 117600, if the numbers are not repeated.

Explanation of Solution

Given:

A combination lock displays 50 numbers. To open it, you turn clockwise to the first number of the “combination,” then rotate counterclockwise to the second number, and then rotate clockwise to the third number.

Formula used:

$C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!},P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

Calculation:

A combination lock displays 50 numbers.

a. If the numbers can be repeated then,

Number of selections for 1st clockwise $=C\left(50,1\right)=50$.

Number of selections for 2nd counter clockwise $=C\left(50,1\right)=50$.

Number of selections for 3rd clockwise $=C\left(50,1\right)=50$.

Hence, the number of different possible is $C\left(50,1\right)*C\left(50,1\right)*C\left(50,1\right)$.

$=50*50*50=125000$

Similarly, if the numbers cannot be repeated then number of different possible is

$P\left(50,3\right)=50*49*48=117600$

To determine

To find:

b. Comment on the description of such a lock as a combination lock.

Answer to Problem 63AYU

b. It can be called as permutation lock.

Explanation of Solution

Given:

A combination lock displays 50 numbers. To open it, you turn clockwise to the first number of the “combination,” then rotate counterclockwise to the second number, and then rotate clockwise to the third number.

Formula used:

$C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!},P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

Calculation:

A combination lock displays 50 numbers.

b. Instead of combination lock it can be called as Permutation lock because, the order of the number is important here.