a 5 digit telephone number is made up of three 1's and two 5's. How many extension numbers of this type can be formed
To find the number of extension numbers that can be formed, we need to consider the arrangements of the three 1's and two 5's in a 5-digit number.
Step 1: Determine the total number of ways to arrange the digits:
Since there are 5 digits in total, we have 5 positions to fill. We can calculate this using the formula for permutations of unique items:
Total arrangements = 5!
Step 2: Account for repetitions of the 1's and 5's:
In this case, we have three 1's and two 5's. The number of arrangements of these repeated digits can be calculated using the formula for permutations of repeated items:
Arrangements of repeated digits = 3! × 2!
Step 3: Find the number of extension numbers:
By dividing the total number of arrangements by the arrangements of repeated digits, we can find the number of extension numbers:
Number of extension numbers = Total arrangements / Arrangements of repeated digits
Therefore, the number of extension numbers that can be formed is:
Number of extension numbers = 5! / (3! × 2!)
Simplifying:
Number of extension numbers = (5 × 4 × 3 × 2 × 1) / [(3 × 2 × 1) × (2 × 1)]
Number of extension numbers = (5 × 4) / 2
Number of extension numbers = 10
So, there are 10 extension numbers of this type that can be formed.