How many different ten digit telephone numbers contain four 2's, three 3s, and three 7s?
I tried sorting the numbers out in order
2 2 2 2 | 3 3 3 | 7 7 7
10!/2!3!7!
But I couldn't find the correct solution as the answer was 4200
no, you probably meant
10!/(4!,3!,3!)
= 4200
Let's say
N=10
a=4 2s
b=3 3s
c=3 7s
Therefore, 10!/(4!×3!×3!)=4200
I hope it helps
To correctly solve this problem, you need to consider the placement of the digits:
1. The four 2's can be placed in the ten-digit number in 10 choose 4 ways: C(10,4).
2. After placing the four 2's, there are six remaining open spots for the three 3's: C(6,3).
3. Finally, after placing the four 2's and three 3's, there are three remaining open spots for the three 7's: C(3,3).
To find the total number of different ten-digit telephone numbers, you need to multiply these three combinations:
C(10,4) * C(6,3) * C(3,3) = 210 * 20 * 1 = 4200.
Therefore, there are 4200 different ten-digit telephone numbers that contain four 2's, three 3's, and three 7's.
To find the number of different ten-digit telephone numbers that contain four 2's, three 3's, and three 7's, you can use a combination of counting principles.
First, let's break down the problem into steps:
Step 1: Determine the positions of the 2's.
Here, we have four 2's to be placed in ten different positions. Think of this as selecting four out of the ten positions for the 2's. This can be calculated using combinations, specifically "10 choose 4," which can be denoted as C(10, 4). The formula is:
C(10, 4) = 10! / (4! * (10 - 4)!) = 10! / (4! * 6!) = 210
Step 2: Determine the positions of the 3's.
Similarly, we have three 3's to be placed in the remaining six positions. This can be calculated as "6 choose 3," which is denoted as C(6, 3). The formula is:
C(6, 3) = 6! / (3! * (6 - 3)!) = 6! / (3! * 3!) = 20
Step 3: Determine the positions of the 7's.
Again, using the same logic, we have three 7's to be placed in the remaining three positions. This can be calculated as "3 choose 3," which is denoted as C(3, 3). The formula is:
C(3, 3) = 3! / (3! * (3 - 3)!) = 1
Step 4: Calculate the total number of arrangements.
The total number of different telephone numbers can be obtained by multiplying the results from each step:
Total number = C(10, 4) * C(6, 3) * C(3, 3) = 210 * 20 * 1 = 4200
Therefore, there are 4200 different ten-digit telephone numbers that contain four 2's, three 3's, and three 7's.