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29 Nov 2024, 03:54
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
62% (01:45) correct
38%
(01:45)
wrong
based on 271
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The elements of set S are all positive integers that can be expressed as the product of two or more of the factors 3, 11, 17 and 23. If no single product can use the same factor twice, how many elements are in Set S?
A 6
B 9
C 11
D 24
E 16
A 6
B 9
C 11
D 24
E 16
29 Nov 2024, 05:02
Kudos
Bookmarks
To determine the number of elements in set S, we need to find all possible products formed by multiplying two or more distinct factors from the set {3, 11, 17, 23}.
1. Identify the Number of Ways to Choose Factors:
• Choosing 2 factors:
The number of ways to choose 2 distinct factors from 4 is calculated using combinations:
C(4, 2) = 6
These combinations are:
(3 × 11), (3 × 17), (3 × 23), (11 × 17), (11 × 23), (17 × 23)
• Choosing 3 factors:
The number of ways to choose 3 distinct factors from 4:
C(4, 3) = 4
These combinations are:
(3 × 11 × 17), (3 × 11 × 23), (3 × 17 × 23), (11 × 17 × 23)
• Choosing all 4 factors:
There’s only one way to choose all 4 factors:
C(4, 4) = 1
Combination:
(3 × 11 × 17 × 23)
2. Calculate the Total Number of Elements:
• Total combinations = C(4, 2) + C(4, 3) + C(4, 4)
• Total combinations = 6 + 4 + 1 = 11
3. Ensure All Products Are Unique:
• Since all factors are distinct prime numbers, each product will be unique. There are no duplicate products.
Conclusion
There are 11 distinct elements in set S.
Option C is the correct answer.
1. Identify the Number of Ways to Choose Factors:
• Choosing 2 factors:
The number of ways to choose 2 distinct factors from 4 is calculated using combinations:
C(4, 2) = 6
These combinations are:
(3 × 11), (3 × 17), (3 × 23), (11 × 17), (11 × 23), (17 × 23)
• Choosing 3 factors:
The number of ways to choose 3 distinct factors from 4:
C(4, 3) = 4
These combinations are:
(3 × 11 × 17), (3 × 11 × 23), (3 × 17 × 23), (11 × 17 × 23)
• Choosing all 4 factors:
There’s only one way to choose all 4 factors:
C(4, 4) = 1
Combination:
(3 × 11 × 17 × 23)
2. Calculate the Total Number of Elements:
• Total combinations = C(4, 2) + C(4, 3) + C(4, 4)
• Total combinations = 6 + 4 + 1 = 11
3. Ensure All Products Are Unique:
• Since all factors are distinct prime numbers, each product will be unique. There are no duplicate products.
Conclusion
There are 11 distinct elements in set S.
Option C is the correct answer.
17 Dec 2024, 01:38
Kudos
Bookmarks
S = {1*3,1*11,1*17,1*23}
Unique Factors are 1,3,11,17,23
Using 1 as factor = 1*3 | 11*1 | 17*1 | 23*1
Product of 2 factors [Excluding 1] = 3*11 | 3* 17 | 3*23 | 11*17| 11*23 | 17*23
Product of 3 factors [Excluding 1] = 3*11*17 | 3*11*23 | 3*17*23 | 11*23*17
Product of 4 factors [Excluding 1] = 3*11*17*23
_ _+ _ _ + _ _ _ + _ _ _ _
4 + 4*3/2! + 4*3*2/3! + 1
4 + 6 + 4 + 1 = 15
Unique Factors are 1,3,11,17,23
Using 1 as factor = 1*3 | 11*1 | 17*1 | 23*1
Product of 2 factors [Excluding 1] = 3*11 | 3* 17 | 3*23 | 11*17| 11*23 | 17*23
Product of 3 factors [Excluding 1] = 3*11*17 | 3*11*23 | 3*17*23 | 11*23*17
Product of 4 factors [Excluding 1] = 3*11*17*23
_ _+ _ _ + _ _ _ + _ _ _ _
4 + 4*3/2! + 4*3*2/3! + 1
4 + 6 + 4 + 1 = 15
tiffanymarcus
The elements of set S are all positive integers that can be expressed as the product of two or more of the factors 3, 11, 17 and 23. If no single product can use the same factor twice, how many elements are in Set S?
A 6
B 9
C 11
D 24
E 16
A 6
B 9
C 11
D 24
E 16
