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The number of ways in which 8 different beads be strung on a necklace is
- a)2500
- b)2520
- c)2250
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The number of ways in which 8 different beads be strung on a necklace ...
The number of ways of arranging n beads in a necklace is (n-1)!/2 = (8-1)!/2 = 7!/2 = 2520
(since n = 8)
Most Upvoted Answer
The number of ways in which 8 different beads be strung on a necklace ...
Given: 8 different beads
To find: Number of ways to string them on a necklace
Solution:
When we string beads on a necklace, we consider circular permutations.
The number of circular permutations of n different objects is (n-1)!.
Therefore, the number of ways to string 8 different beads on a necklace is (8-1)! = 7!.
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
However, since the necklace can be rotated, we need to divide by 8 to eliminate duplicate arrangements.
Therefore, the number of distinct ways to string the beads on a necklace is 5040/8 = 630.
Option (b) is the correct answer: 2520, which is the number of ways to arrange the beads on the necklace when reflections are not considered. However, since reflections are allowed in this problem, we need to divide by 2 to get the correct answer of 2520/2 = 1260.
To find: Number of ways to string them on a necklace
Solution:
When we string beads on a necklace, we consider circular permutations.
The number of circular permutations of n different objects is (n-1)!.
Therefore, the number of ways to string 8 different beads on a necklace is (8-1)! = 7!.
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
However, since the necklace can be rotated, we need to divide by 8 to eliminate duplicate arrangements.
Therefore, the number of distinct ways to string the beads on a necklace is 5040/8 = 630.
Option (b) is the correct answer: 2520, which is the number of ways to arrange the beads on the necklace when reflections are not considered. However, since reflections are allowed in this problem, we need to divide by 2 to get the correct answer of 2520/2 = 1260.
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Community Answer
The number of ways in which 8 different beads be strung on a necklace ...
We are asked to find the number of ways to string 8 different beads on a necklace.
Step 1: Understanding the Problem
When arranging beads in a circular pattern, we must account for rotational symmetry (i.e., rotating the arrangement does not create a new arrangement). Additionally, for a necklace, we must also account for reflection symmetry (i.e., flipping the arrangement over does not create a new arrangement).
When arranging beads in a circular pattern, we must account for rotational symmetry (i.e., rotating the arrangement does not create a new arrangement). Additionally, for a necklace, we must also account for reflection symmetry (i.e., flipping the arrangement over does not create a new arrangement).
Step 2: Calculating the Number of Ways
The formula to calculate the number of ways to arrange n different beads on a necklace is:
This formula accounts for both rotational and reflection symmetries.
The formula to calculate the number of ways to arrange n different beads on a necklace is:
This formula accounts for both rotational and reflection symmetries.
For n = 8 (since we have 8 different beads), we can substitute the value of n into the formula:
Step 3: Calculating the Factorial
The factorial of 7 (7!) is:
7! = 7 x 6 x5 x4 x 3 x2x1 = 5040
Now, divide by 2 to account for reflection symmetry:
5040 / 2 = 2520
Step 3: Calculating the Factorial
The factorial of 7 (7!) is:
7! = 7 x 6 x5 x4 x 3 x2x1 = 5040
Now, divide by 2 to account for reflection symmetry:
5040 / 2 = 2520
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The number of ways in which 8 different beads be strung on a necklace isa)2500b)2520c)2250d)none of theseCorrect answer is option 'B'. Can you explain this answer?
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