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.
Well, it sounds like Mary wants a ring that really rocks! Let me break it down for you:
Part A: To find the number of ways the three stones can be arranged, we use the concept of permutations. Since order matters, we multiply the number of choices for each position. In this case, there are 12 choices for the first stone, 11 choices for the second stone (after the first stone has been selected), and 10 choices for the third stone (after the first two stones have been selected). So the expression to show how many ways the stones can be arranged is 12 * 11 * 10. To evaluate it, we simply multiply these numbers together: 12 * 11 * 10 = 1320.
Part B: To find the number of permutations with 4 different stones, we again use the concept of permutations. Since order matters, we multiply the number of choices for each position. In this case, there are 12 choices for the first stone, 11 choices for the second stone, 10 choices for the third stone, and 9 choices for the fourth stone. So the expression to show how many permutations are possible is 12 * 11 * 10 * 9. Evaluating this expression gives us: 12 * 11 * 10 * 9 = 11,880.
Part C: Ah, the eternal question of 4 stones versus 5 stones! Well, to compare the number of rings that can be made, we need to find the number of permutations possible for 5 different stones. Using the same logic as before, there are 12 choices for the first stone, 11 choices for the second stone, 10 choices for the third stone, 9 choices for the fourth stone, and 8 choices for the fifth stone. So the expression to show how many permutations are possible with 5 different stones is 12 * 11 * 10 * 9 * 8. Evaluating this expression gives us: 12 * 11 * 10 * 9 * 8 = 95,040. As you can see, there are more permutations possible with 5 different stones compared to 4 different stones, which means more rings can be made with 5 different stones. Quite the gem of a discovery, isn't it?
Part A: To calculate the number of ways the 3 stones can be arranged in a line, we can use the concept of permutations. The formula for calculating permutations is nPr = n! / (n-r)!, where n is the total number of items and r is the number of items selected at a time.
In this case, we have 12 different birthstones to choose from, and we need to select 3 stones. Therefore, the expression to show how many ways the 3 stones can be arranged is:
12P3 = 12! / (12-3)! = 12! / 9! = (12 * 11 * 10) / (3 * 2 * 1) = 220
So, there are 220 ways to arrange the 3 stones.
Part B: To calculate the number of permutations possible with 4 different stones of any month, we again use the permutation formula.
In this case, we have 12 different birthstones to choose from, and we need to select 4 stones. Therefore, the expression to calculate the number of permutations is:
12P4 = 12! / (12-4)! = 12! / 8! = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1) = 495
So, there are 495 permutations possible with 4 different stones.
Part C: To determine if more rings can be made with 5 different stones than with 4 different stones, we compare the number of permutations for each case.
From Part B, we know that the number of permutations with 4 different stones is 495.
To calculate the number of permutations with 5 different stones, we use the same formula:
12P5 = 12! / (12-5)! = 12! / 7! = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1) = 792
So, there are 792 permutations possible with 5 different stones.
Since 792 is greater than 495, it means that more rings can be made with 5 different stones than with 4 different stones. This is because having one additional stone increases the number of possible combinations.
Part A:
To calculate the number of ways the 3 stones can be arranged, we can use the concept of permutations.
Permutations are the different ways in which a set of objects can be arranged or ordered. In this case, we are arranging the 3 stones.
The formula for permutations is P(n, r) = n! / (n - r)!, where n is the total number of objects and r is the number of objects being arranged.
So, in this case, we have 12 stones (one for each month) and want to arrange 3 of them. Therefore, the expression for Part A would be:
P(12, 3) = 12! / (12 - 3)!
= 12! / 9!
= (12 x 11 x 10 x 9!) / 9!
= 12 x 11 x 10
= 1,320
So, there are 1,320 different ways in which the 3 stones can be arranged.
Part B:
To calculate the number of permutations possible with 4 different stones of any month, we can again use the formula for permutations.
In this case, we have 12 stones and want to arrange 4 of them. Therefore, the expression for Part B would be:
P(12, 4) = 12! / (12 - 4)!
= 12! / 8!
= (12 x 11 x 10 x 9 x 8!) / 8!
= 12 x 11 x 10 x 9
= 11,880
So, there are 11,880 different permutations possible with 4 different stones.
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 know that there are 11,880 permutations possible with 4 different stones.
To calculate the number of permutations possible with 5 different stones, we would use the expression:
P(12, 5) = 12! / (12 - 5)!
= 12! / 7!
= (12 x 11 x 10 x 9 x 8 x 7!) / 7!
= 12 x 11 x 10 x 9 x 8
= 95,040
So, there are 95,040 different permutations possible with 5 different stones.
Therefore, more rings can be made with 5 different stones (95,040) than with 4 different stones (11,880). The additional stone provides more options for arrangement, resulting in a greater number of possible rings.
A)3!=6 ways B)4!=24 ways C)5!=120 ways ...since there is 12 to choose from.
That's on my test too!!!