Video Transcript
7 digit number and the very first digit cannot be 0 or 1. So that means your very first digit could be a total of 8 different possibilities, because our possibilities are 012345678 and 9, but 0 and 1 are not possible, so that leaves 8 possibilities left. If you just count these there's 8, so that's 8 possibilities for a first digit now, every other digit could be anything between 0 and 9. So if you count that up that's 10 possibilities, so these could all be 10 possible outcomes and then you just multiply to find the total number of outcomes. So let's multiply this and we get a total 8000000 sorry about that 8000000. Now the next part is we're doing this again, except now, we're also going to assume the phone number cannot be 911 to start. So again we have a 7 digit number, the first 2 digits, i'm sorry. The first digit again cannot be 0 or 1, but now we can't have 9 either so for your very first digit, it can't be 0 or 1 or 9, so that leaves 7 possibilities. Now the second digit can't be a 1, but it could be any of the other 9 possibilities. The third digit cannot be a 1 either, but it could be any of the other 9 possibilities now for the last 4 digits. They could be anything so any of the 10 possibilities are fine, so now multiplying these together, we end up with 5076600 total different outcomes.
Answer
Verified
Hint: Here, first find the total number of ways where the first digit cannot be either 1 or 0. Now, find the total number of ways where the first three-digits can be either 911 or 411. Finally, subtract the number of ways with the second condition from the number of ways with the first condition to find the required number of ways.Complete step-by-step solution:
Here, we have two conditions, one of those conditions is that the seven-digit number which is the area
code cannot be either 1 or 0. Also, the other condition is that the first three-digits cannot be 911 and 411.
So, let us first take the condition of the first digit to not be either 1 and 0.
We need to find the seven-digit number, which is given
Here, the above table represents the form of the seven-digit number, now we have numbers, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
In the first box, we cannot fill it with 1 and 0, and hence, we can fill the 1st box with 8 remaining possible
numbers. Similarly, we can fill the rest of the boxes by all the ten numbers according to the first condition.
Therefore, according to the fundamental principle of multiplication, we get
Total number of ways seven-digit number can be arranged = 8 x 10 x 10 x 10 x 10 x 10 x 10
=
8,000,000.
Therefore, according to the condition that the first digit cannot be either 1 or 0, we can have 8,000,000 possible ways of creating a seven-digit area code.
Now, according to the other condition which states that the first three-digits cannot have the numbers 911 or 411.
For 911:
According to the fundamental principle of multiplication,
Total number of ways = 1 x 1 x 1 x 10 x 10 x 10 x 10
=
10,000
For 411:
According to the fundamental principle of multiplication.
Total number of ways = 1 x 1 x 1 x 10 x 10 x 10 x 10
= 10,000
Therefore, total number of ways for 911 and 411 = 10,000 + 10,000
=
20,000
Here, 20,000 possible ways to arrange the seven-digit number with the first three digits as either 911 or 411.
Now, the condition says that the seven-digit number cannot have 911 and 411 as the three digits. Hence, we need to subtract 20,000 from 8,000,000 which will give us the total number of ways satisfying both the conditions mentioned in the question.
Total required number of ways for the seven-digit number = 8,000,000 – 20,000
=
7,980,000
Hence, the seven-digit area code can be obtained in 7,980,000 possible ways.
Note: The fundamental principle of multiplication states that, if an operation can be performed in ‘m’ different ways, following which is the second operation can be performed in ‘n’ different ways, then the two operations in succession can be performed in ‘m x n’ ways.