(a)Sarah wants to make a bracelet. She needs some beads and goes to
a shop where they sell beads in exactly 100 different colours.
i. How many ways can she pick 25 beads, each of a different colour?
ii. How many ways can she pick any 25 beads?
(b) Suppose that Sarah has picked 25 beads each of different colours.
i If Sarah arranges all of these beads in a line on the table in
front of her how many different arrangements are possible? What
about if she arranges them in a circle on the table, how many
arrangements are possible if two arrangements are considered
identical if they can be obtained from one another by rotation?
ii. If Sarah uses all of the 25 beads to make her bracelet, how many
distinct bracelets can she make ?
iii. What if she decides to make her bracelet using at least 20 of the
beads, how many distinct bracelets can she make then?
(a) To solve these problems, we can use the concept of combinations and permutations.
i. To pick 25 beads, each of a different color, we need to select one bead of each color. Since there are 100 different colors available, we can choose the first bead in 100 ways, the second bead in 99 ways, the third bead in 98 ways, and so on until the 25th bead, which we can choose in 76 ways. The total number of ways to pick 25 beads, each of a different color, is calculated as:
100 * 99 * 98 * ... * 76 = 100! / (100 - 25)! = 3.5 x 10^35
ii. To pick any 25 beads regardless of their colors, we can choose any bead for each of the 25 positions. Since there are 100 different colors available for each position, the total number of ways to pick any 25 beads is:
100^25 = 10^50
(b) Now let's consider the arrangements of the beads.
i. If Sarah arranges all of the 25 beads in a line, the number of different arrangements is given by the factorial of 25 (25!), which is equal to:
25! = 1 * 2 * 3 * ... * 23 * 24 * 25 = 1.5 x 10^25
If she arranges them in a circle, we need to account for the fact that two arrangements are considered identical if they can be obtained from each other by rotation. There are 25 possible rotations for each arrangement. Therefore, the number of different arrangements in a circle is given by:
25! / 25 = 24!
ii. If Sarah uses all 25 beads to make her bracelet, she needs to arrange them in a circle. As mentioned before, the number of different arrangements in a circle is 24! (as one rotation is removed as an identical arrangement). Therefore, she can make:
24! = 6.2 x 10^23
distinct bracelets.
iii. If Sarah decides to make her bracelet using at least 20 of the beads, we can calculate the number of distinct bracelets by considering the different combinations of the beads. Starting from 20 beads, she can choose up to 25 beads. We can calculate this by summing the possibilities for each number of beads from 20 to 25:
24! + 23! + 22! + 21! + 20! = 6.3 x 10^23 + 6.7 x 10^22 + 1.1 x 10^21 + 2.4 x 10^19 + 2.4 x 10^17
Therefore, she can make approximately:
6.3 x 10^23 distinct bracelets using at least 20 beads.