Question

Within a given area code, calculate the total number of seven-digit numbers available, given the restrictions that the first three-digits cannot be 911 or 411 and the first digit cannot be a 1 or a 0.
(a) 604,800
(b) 2,893,401
(c) 6,380,000
(d) 7,980,000
(e) 8,000,000

Answer 1

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

1st

2nd

3rd

4th

5th

6th

7th

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.