a set of five positive integers has a mean median and range of 7 how many distant sets whose members are listed from least to greatest could have these properties
You missed one! you could also count { 4, 6, 7, 7, 11}.
Well, let's put on our math hats and clown around with this problem!
To find the possible sets, let's call the integers a, b, c, d, and e, listed from least to greatest. We know that the mean, median, and range are all 7.
The mean of the set is the sum of the numbers divided by their count. So, we have a + b + c + d + e = 5 * 7 = 35.
Since the median is the middle number when the set is sorted, the third number must be 7. So, we have a, b, 7, d, and e.
The range is the difference between the greatest and least numbers. Since the range is 7, we have e - a = 7.
Now, since e > a, we know that e is at least 8. This means the maximum value for e is 15, since e + 7 = a + 14.
To find the number of distant sets, we need to consider all possible values for e.
If e = 8, then a = 1. The set would be {1, b, 7, d, 8}.
If e = 9, then a = 2. The set would be {2, b, 7, d, 9}.
Continuing this pattern, if e = 15, then a = 8. The set would be {8, b, 7, d, 15}.
So, we have a total of 15 - 8 + 1 = 8 distant sets whose members are listed from least to greatest and have the given properties.
Remember, math can be fun, especially when you add a little clowning around!
To find the number of distinct sets with the given properties, we need to determine the possible values for the set.
Let's denote the five positive integers in the set as a, b, c, d, and e (listed from least to greatest).
Mean:
The mean is the average of the numbers in the set. In this case, it is given as 7. So, we have:
(a + b + c + d + e) / 5 = 7
Median:
The median is the middle value when the numbers are arranged in ascending order. Since we have an odd number of elements, the median will be the third element. So, we can write:
c = 7
Range:
The range is the difference between the largest and smallest values in the set. Since the range is given as 7, we have:
e - a = 7
From the given conditions, we have three equations:
(a + b + d + e) = 28 -- (equation 1)
c = 7 -- (equation 2)
e - a = 7 -- (equation 3)
Now, let's try to find the distinct sets that satisfy these properties.
Since c = 7, the third number in the set will always be 7. We can arrange the remaining four variables (a, b, d, and e) in different ways respecting the constraints given by equation 1 and equation 3.
Let's consider the minimum and maximum values for a and e:
Minimum values:
Since a, b, d, and e are positive integers, the minimum values we can choose are:
a = 1, e = 8
Maximum values:
The maximum value for a would be such that a + b + d + e = 28. Since b, d, and e must also be positive integers, the maximum value for a would be:
a = 20, b = d = e = 2
Now, we have obtained the minimum and maximum values of a and e. To find the number of distinct sets, we can count the number of values between these minimum and maximum values.
The possible values for a range from 1 to 20 (inclusive) and for e, it ranges from 8 to 2. Hence, to get the number of distinct sets, we count the number of values between 1 and 20 (inclusive) for a and multiply it by the number of values between 8 and 2 (inclusive) for e.
Therefore, the number of distinct sets that satisfy the given conditions is: (20 - 1 + 1) * (8 - 2 + 1) = 120.
So, there are 120 distinct sets whose members are listed from least to greatest and have a mean, median, and range of 7.
a+b+c+d+e = 35 (mean=7)
c = 7 (median)
e = a+7 (range)
a+b+7+d+a+7 = 35
2a+b+d = 21
If a=1, b+d=19
But b<=7 and e=8
1,b,7,d,8
But, if b<=7, and d<=8, no go
If a=2, b+d=17
2,b,7,d,9
same problem as above
If a=3, b+d=15
d<=10, so b>=5
so we can have
3,b,7,d,10 as
3,5,7,10,10
3,6,7,9,10
3,7,7,8,10
If a=4, b+d=13
d <= 11, so b>=2. In fact, we know b>=4, so we can have
4,b,7,d,11
4,4,7,9,11
4,5,7,8,11
If a=5, b+d=11
d<=12,
5,b,7,d,12
no go, since b>=5, meaning d<=6, but d must be at least 7.
So, only the sets in boldface above are candidates.