A store sells rings with birthstones selected from the 12 different months of the year. The stones are arranged in a row.
Part A: Mary wants a ring with a topaz, a sapphire, and a ruby set in a line. Write and evaluate an expression to show how many ways the 3 stones can be arranged?
Part B: How many permutations are possible with 4 different stones of any month?
Part C: Can more rings be made with 5 different stones than with 4 different stones? Explain.
Part A:
Assuming the ring is not symmetrical, i.e. it cannot be worn both ways, then there are 3! ways the stones can be arranged.
If the ring is symmetrical, divide by 2.
Part B: similar to A, replace 3 by 4.
Part C: Leave it to you as an exercise.
Part A: To find the number of ways the 3 stones can be arranged, we need to use the concept of permutations. The number of permutations of n distinct objects, taken r at a time, is given by nPr, which is the formula for permutations.
In this case, we have 3 stones (topaz, sapphire, and ruby) that need to be arranged. So the expression to show how many ways the stones can be arranged is 3P3.
Evaluating this expression:
3P3 = 3! / (3-3)! = 3! / 0! = 3! = 3 x 2 x 1 = 6
Therefore, there are 6 ways the topaz, sapphire, and ruby stones can be arranged.
Part B: To find the number of permutations possible with 4 different stones of any month, we again use the formula nPr. In this case, n (number of stones) is 4 and r (number of stones to be arranged) is also 4.
So the expression to show how many permutations are possible is 4P4.
Evaluating this expression:
4P4 = 4! / (4-4)! = 4! / 0! = 4! = 4 x 3 x 2 x 1 = 24
Therefore, there are 24 permutations possible with 4 different stones of any month.
Part C: No, more rings cannot be made with 5 different stones than with 4 different stones. The number of permutations increases as the number of elements to be arranged increases. In this case, the number of permutations of 4 different stones (24) is greater than the number of permutations of 5 different stones.
Part A: To find the number of ways the 3 stones can be arranged, we can use the concept of permutations. Since there are 3 stones to arrange, we can calculate the number of permutations using the formula:
nPr = n! / (n-r)!
In this case, n represents the total number of stones (12), and r represents the number of stones we want to select and arrange (3). Plugging these values into the formula, we get:
12P3 = 12! / (12-3)!
= 12! / 9!
Evaluating this expression:
12! = 12 × 11 × 10 × 9!
= 12 × 11 × 10
So, the expression becomes:
12P3 = (12 × 11 × 10) / 9!
= 1320 / 9!
= 1320 / (9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)
= 1320 / 362880
Hence, there are 1320 different ways to arrange 3 stones selected out of 12 birthstones.
Part B: To find the number of permutations with 4 different stones, we can again use the formula for permutations:
nPr = n! / (n-r)!
Here, n is the total number of stones (12), and r is the number of stones we want to select and arrange (4). Plugging these values into the formula, we get:
12P4 = 12! / (12-4)!
= 12! / 8!
Evaluating this expression:
12! = 12 × 11 × 10 × 9 × 8!
= 12 × 11 × 10 × 9
So, the expression becomes:
12P4 = (12 × 11 × 10 × 9) / 8!
= 11,880 / (8 × 7 × 6 × 5 × 4 × 3 × 2 × 1)
= 11,880 / 40,320
Hence, there are 11,880 different permutations possible with 4 different stones of any month.
Part C: To determine if more rings can be made with 5 different stones than with 4 different stones, we need to compare the number of permutations for each case.
From Part B, we found that the number of permutations with 4 different stones is 11,880.
To calculate the number of permutations with 5 different stones, we again use the formula for permutations:
12P5 = 12! / (12-5)!
= 12! / 7!
Evaluating this expression:
12! = 12 × 11 × 10 × 9 × 8 × 7!
= 12 × 11 × 10 × 9 × 8 × 7
So, the expression becomes:
12P5 = (12 × 11 × 10 × 9 × 8 × 7) / 7!
= 79,380 / (7 × 6 × 5 × 4 × 3 × 2 × 1)
= 79,380 / 5,040
Hence, there are 79,380 different permutations possible with 5 different stones.
Since 79,380 is larger than 11,880, it means that more rings can be made with 5 different stones than with 4 different stones.