Evaluate the expression.
Determine the number of combinations of 7 things taken 4 at a time.
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Mark can remember only the first 4 digits of his friend's phone number. He also knows that the number has 7 digits and that the last digit is not a 0. If Mark were to dial all of the possible numbers and if it takes him 21 seconds to try each one, how long would it take to try every possibility?
7C4 = 7C3 = 7*6*5 / 1*2*3 = 35
The number of choices is 10*10*9
10*10*9 * 21 = 18,900 seconds
That's only 5 hr 15 min
To evaluate the expression for the number of combinations of 7 things taken 4 at a time, we can use the formula for combinations:
C(n, r) = n! / (r! * (n-r)!)
where n is the total number of things and r is the number of things taken at a time.
In this case, we have 7 things and we are taking 4 at a time, so n = 7 and r = 4.
Substituting these values into the formula, we get:
C(7, 4) = 7! / (4! * (7-4)!)
= 7! / (4! * 3!)
= (7 * 6 * 5 * 4!) / (4! * 3 * 2 * 1)
= (7 * 6 * 5) / (3 * 2 * 1)
= (7 * 6 * 5) / 6
= 35
So, the number of combinations of 7 things taken 4 at a time is 35.
Now, let's calculate the time it would take for Mark to try every possible number.
We know that the number has 7 digits and the last digit is not a 0. So, the last digit has 9 possible options (1 to 9). The remaining 6 digits can take any of the 10 possible values (0 to 9).
Therefore, the total number of possible numbers is: 9 x 10 x 10 x 10 x 10 x 10 x 10 = 9 x 10^6
Since Mark takes 21 seconds to try each number, the total time it would take to try every possibility is:
9 x 10^6 x 21 seconds
= 189 x 10^6 seconds
= 189,000,000 seconds
So, it would take 189,000,000 seconds to try every possible number.