(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)
i. To find the number of ways Sarah can pick 25 beads, each of a different color, we can use the concept of combinations. Since there are 100 different colors of beads, she can choose one color out of the 100 for the first bead, then choose another color out of the remaining 99 for the second bead, and so on. The number of ways to choose 25 beads, each of a different color, can be calculated using the formula for combinations:
nCr = n! / (r! * (n-r)!)
Where n is the total number of colors (100) and r is the number of beads to be chosen (25).
Using the formula, we can calculate:
100C25 = 100! / (25! * (100-25)!)
ii. To find the number of ways Sarah can pick any 25 beads, it means she can choose multiple beads of the same color. In this case, it becomes a case of permutations because the order in which the beads are chosen matters. The formula for permutations is:
nPr = n! / (n-r)!
Where n is the total number of colors (100) and r is the number of beads to be chosen (25).
Using the formula, we can calculate:
100P25 = 100! / (100-25)!
(b)
i. If Sarah arranges all the 25 beads in a line on the table, the number of different arrangements can be calculated using the formula for permutations, as the order of beads matters.
25P25 = 25!
ii. If Sarah arranges the beads in a circle on the table, we need to consider that two arrangements are considered identical if they can be obtained from one another by rotation. In this case, the number of arrangements is reduced by a factor of 25. Hence, the number of different arrangements in a circle can be calculated using the formula:
(25P25) / 25
iii. If Sarah uses all 25 beads to make her bracelet but allows at least 20 of the beads to be used, we can calculate the number of distinct bracelets by summing up the possible cases for using 20, 21, 22, 23, 24, and 25 beads. For each case, we can use the formula as mentioned above to calculate the number of different arrangements.
The total number of distinct bracelets = [(20P20) / 20] + [(21P21) / 21] + [(22P22) / 22] + [(23P23) / 23] + [(24P24) / 24] + [(25P25) / 25]
The individual terms represent the number of distinct bracelets when using exactly 20, 21, 22, 23, 24, and 25 beads respectively.