How many different seven-digit telephone numbers are available if the only restriction is that the first digit can't be 0? Is there a formula for this to make it simpler? Then I'll solve this problem myself.
there are 9 ways to fill the first place
there are 10 ways to fill the second place, so far we have 9x10
there are 10 ways to fill the third place, so far we 9x10x10
etc.
9,000,000
To find the answer to this question, we need to understand the concept of permutations. The number of different seven-digit telephone numbers, with the first digit not being 0, can be calculated using permutations.
Permutations represent the different arrangements or orders in which a set of objects can be arranged.
In this case, we have a set of 10 possible digits (0 to 9) for each digit position, with the restriction that the first digit cannot be 0. We can think of it as having 9 options for the first digit and 10 options for each of the remaining 6 digits.
To calculate the total number of seven-digit telephone numbers, we can multiply the number of options for each digit position together.
The formula for calculating permutations in this case is:
P(n, r) = n × (n-1) × (n-2) × ... × (n-r+1)
where n is the total number of options and r is the number of positions to be filled.
For our problem, n = 10 (since we have 10 digits) and r = 7 (since we have 7 positions to be filled).
Therefore, the formula becomes:
P(10, 7) = 10 × 9 × 8 × 7 × 6 × 5 × 4
Simplifying this gives us:
P(10, 7) = 3,628,800
So, there are 3,628,800 different seven-digit telephone numbers available if the only restriction is that the first digit cannot be 0.
Now that you know the formula, you can verify the solution yourself or apply it to similar problems.