a company makes combination locks with 50 numbers printed on the dial. Each lock combination is an arrangement of 3 different numbers. How many locks can the company make without repeating a combination?
A Permutation is an ordered Combination.
There are basically two types of permutation:
1-Repetition is Allowed.
2-No Repetition is allowed.
Your question is one without repetition.
To answer, multiply 50 x 49 x 48 =???
Why did I stop at 48?
Because the question states
"Each lock combination is an arrangement of 3 different numbers."
Now multiply the above numbers and you will find the answer.
Done!
To find out how many locks the company can make without repeating a combination, we need to determine the number of possible combinations of 3 different numbers chosen from a set of 50 numbers.
The mathematical concept that relates to this problem is called "combination." The formula to calculate the number of combinations of selecting r objects from a set of n objects is given by:
C(n, r) = n! / (r! * (n - r)!)
Here, n is the total number of available options (50 numbers on the dial) and r is the number of objects to be chosen (3).
Using this formula, we can calculate the number of combinations:
C(50, 3) = 50! / (3! * (50 - 3)!)
Now let's calculate it step-by-step:
Step 1: Calculate 50! (factorial of 50)
50! = 50 * 49 * 48 * ... * 3 * 2 * 1
Step 2: Calculate 3! (factorial of 3)
3! = 3 * 2 * 1
Step 3: Calculate (50 - 3)! (factorial of 47)
47! = 47 * 46 * ... * 3 * 2 * 1
Step 4: Substitute these values into the combination formula:
C(50, 3) = 50! / (3! * (50 - 3)!)
= (50 * 49 * 48 * ... * 3 * 2 * 1) / (3 * 2 * 1) * (47 * 46 * ... * 3 * 2 * 1)
Step 5: Simplify the equation:
C(50, 3) = (50 * 49 * 48) / (3 * 2 * 1) * (47 * 46 * ... * 3 * 2 * 1)
Step 6: Cancel out common factors:
C(50, 3) = (50 * 49 * 48) / (3 * 2 * 1) * (47 * 46 * ... * 4)
Now we can calculate the value:
50 * 49 * 48 = 117,600
3 * 2 * 1 = 6
47 * 46 * ... * 4 = 103,776
C(50, 3) = (117,600) / (6) * (103,776) = 19,600,400
Therefore, the company can make 19,600,400 locks without repeating a combination.
To determine the number of locks the company can make without repeating a combination, we need to find the number of ways to choose 3 different numbers out of 50 without repetition.
To solve this, we can use the concept of combinations. The formula for finding the number of combinations is given by:
C(n, r) = n! / (r!(n - r)!)
Where:
- C(n, r) denotes the number of combinations of choosing r items from a set of n items.
- n! represents n factorial, which is the product of all positive integers from 1 up to n.
In this case, we have to find C(50, 3). Plugging in the values into the formula, we get:
C(50, 3) = 50! / (3!(50 - 3)!)
Simplifying further:
C(50, 3) = 50! / (3!47!)
Now, we can calculate the factorials:
50! = 50 × 49 × 48 × ... × 3 × 2 × 1
3! = 3 × 2 × 1
47! = 47 × 46 × 45 × ... × 3 × 2 × 1
By canceling out common terms in the numerator and denominator:
C(50, 3) = (50 × 49 × 48) / (3 × 2 × 1)
Performing the multiplication and division:
C(50, 3) = 19,600
Therefore, the company can make 19,600 different locks without repeating any combination.