a Make sure you list all of the possible permutations of the three randomly selected integers.
B
b Repeat the process from Part A, remembering that each time there should be 6 possible permutations.
C
c What can you say about the average of the permutations looking at the table from Part B?
D
d Notice that each three-digit number abc can be rewritten as the sum 100a+10b+c.
A
a See solution.
B
bExample Solution:
Integers
Permutations
Average of Permutations
Average of Permutations/37
1,4,7
147,174,417,471,714,741
444
12
0,6,3
063,036,603,630,306,360
333
9
2,5,8
258,285,528,582,825,852
555
15
9,7,1
971,917,791,719,197,179
629
17
4,0,2
402,420,042,024,240,204
222
6
C
c The value of the average of the permutations of the three digits between 0 and 9 divided by 37 is equal to the sum of the digits.
D
d A=37(x+y+z)
Practice makes perfect
a We are asked to randomly select 3 digits from the following integers and find the possible permutations of these 3 numbers.
{ 0,1,2,3,4,5,6,7,8,9}
Let's choose 0, 3, and 6. Now, we will find and list all of the possible permutations of these three digits.
0 3 6, 0 6 3, 6 0 3, 6 3 0, 3 0 6, 3 6 0
By the Fundamental Counting Principle we know that there should be 3*2*1=6 possible permutations of 3 digits. We listed all of the possible permutations.
b Now, we want to repeat the process from Part A three more times. Let's choose 3 random integers, list all of the permutations, calculate the average of permutations, and finally divide the average by 37. Notice that in the case of arrangements of 3 objects we have 6 possible permutations.
Integers
Permutations
Average of Permutations
Average of Permutations/37
1, 4, 7
1 4 7, 1 7 4, 4 1 7, 4 7 1, 7 1 4, 7 4 1
444
12
0, 6, 3
0 6 3, 0 3 6, 6 0 3, 6 3 0, 3 0 6, 3 6 0
333
9
2, 5, 8
2 5 8, 2 8 5, 5 2 8, 5 8 2, 8 2 5, 8 5 2
555
15
9, 7, 1
9 7 1, 9 1 7, 7 9 1, 7 1 9, 1 9 7, 1 7 9
629
17
4, 0, 2
4 0 2, 4 2 0, 0 4 2, 0 2 4, 2 4 0, 2 0 4
222
6
c Looking at the table we can see that the average of permutations made from 3 integers divided by 37 is equal to the sum of the integers.
Integers
Sum of the Integers
Average of Permutations/37
1,4,7
1+4+7= 12
12
0,6,3
0+6+3= 9
9
2,5,8
2+5+8= 15
15
9,7,1
9+7+1= 17
17
4,0,2
4+0+2= 6
6
We should examine the formula for the average of permutations further to find out why we can observe this relationship when we divide the average by 37.
d This time we are asked to determine if we can write an equation for the average A of the permutations of the three digits x, y, and z. To do this let's start by listing all of the possible permutations. Just as before, there will be 6 such arrangements.
x y z, x z y, y x z, y z x, z x y, z y x
Now, the average will be the sum of all of the three-digit numbers divided by 6.
A = xyz + xzy + yxz + yzx + zxy + zyx/6
Each three-digit number can be rewritten as the sum. To do so, we need to multiply the first digit by 100 because it represents hundreds, the second digit — representing tens — by 10, and the third digit by 1, as it represents units. Let's use this information to simplify the expressions in the numerator.
Expression
Rewrite as Sum
x y z
100 x+10 y+ z
x z y
100 x+10 z+ y
y x z
100 y+10 x+ z
y z x
100 y+10 z+ x
z x y
100 z+10 x+ y
z y x
100 z+10 y+ x
Let's substitute the sums for the expressions in the numerator. Then we will be able to combine like terms.
The average of the permutations of the three digits can be expressed as the sum of these digits multiplied by 37. This is the reason why after dividing the average of the permutations by 37 in a table from Part B we were ending with the sum of the integers.
