How many ten-digit telephone numbers are possible if the first three digits must be different? The answer is 72,000,000,000.... I need the method to get that:)
They must assume that the lead digit could be a zero
so first 3 digits : 10 x 9 x 8 = 720
That leaves 7 more digits where anything goes
that would be 10^7
so final number = 720 x 10^7
= 7 200 000 000
I disagree with their answer.
Ohh i accidently added an extra zero...thank you:)
To find the number of ten-digit telephone numbers possible, given that the first three digits must be different, we can break down the problem step-by-step:
Step 1: Count the number of possibilities for each digit.
- For the first digit, any number from 1 to 9 can be chosen, as the first digit cannot be zero. So, there are 9 possibilities.
- For the second digit, any number from 0 to 9 can be chosen, except the number already chosen for the first digit. So there are 9 possibilities.
- For the third digit, any number from 0 to 9 can be chosen, except the two numbers already chosen for the first two digits. So there are 8 possibilities.
Step 2: Determine the number of possibilities for the remaining seven digits.
- For each of the remaining seven digits (fourth to tenth), any number from 0 to 9 can be chosen, as no restrictions are given. So there are 10 possibilities for each digit.
Step 3: Calculate the total number of possibilities by multiplying all the previous results.
- Multiply the number of possibilities for each step: 9 × 9 × 8 × 10 × 10 × 10 × 10 × 10 × 10 × 10.
- The result is 72,000,000,000, which corresponds to 72 billion telephone numbers.
Therefore, the method to get 72,000,000,000 possible telephone numbers is to multiply the different possibilities for each digit.
To find the number of ten-digit telephone numbers possible with the first three digits being different, we can break down the problem step by step:
Step 1: Count the number of options for the first digit. Since it cannot be zero, we have 9 choices (1-9).
Step 2: Count the number of options for the second digit. Since it can be any digit except for the one already chosen in the first step, we have 9 choices remaining.
Step 3: Count the number of options for the third digit. Similar to the second digit, it can be any digit except for the two already chosen in the first two steps. So we have 8 choices remaining.
Step 4: Determine the number of options for the remaining seven digits. Each digit can be any number from 0-9, so we have 10 choices for each remaining digit.
To calculate the total number of possibilities, we multiply the number of options for each step together:
Total number of possibilities = (Number of options for the first digit) × (Number of options for the second digit) × (Number of options for the third digit) × (Number of options for the remaining seven digits)
Total number of possibilities = 9 × 9 × 8 × 10^7
Simplifying this equation, we get:
Total number of possibilities = 72,000,000,000
Therefore, there are 72 billion possible ten-digit telephone numbers when the first three digits must be different.