Suppose the ice cream shop offers 9 different flavours, and we share 5 scoops. How many ways of ordering are there if
All different
any combination
two scoops of one flavour, two scoops of 2nd, and one of third
first one, i did 9! / 5!?
second one and third one idk
To calculate the number of ways of ordering, we can use combinations and permutations.
1) All different:
In this case, we have 9 flavors and we want to choose 5 scoops. Since the order matters (e.g., choosing chocolate first or last), we need to use permutations. We can calculate this using the formula for permutations by dividing the total number of possible permutations by the number of repetitions. The formula is nPr = n! / (n - r)!, where n is the total number of items and r is the number of items to be chosen. So, for this scenario, the calculation would be: 9P5 = (9!)/(9-5)! = (9!)/(4!) = 9 x 8 x 7 x 6 x 5 = 15,120 ways of ordering.
2) Any combination:
In this case, we don't care about the order in which the scoops are chosen. We simply want to count the number of combinations. We can calculate this using the formula for combinations. The formula is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items to be chosen. So, for this scenario, the calculation would be: 9C5 = (9!)/(5!(9-5)!) = (9!)/(5!4!) = 9 x 8 x 7 x 6 x 5 / (5 x 4 x 3 x 2 x 1) = 126 ways of ordering.
3) Two scoops of one flavor, two scoops of the second flavor, and one of the third:
In this case, we need to consider the different flavors and their combinations. We have 9 flavors, so we need to consider choosing 2 out of 9 flavors, and then 2 out of the remaining 8 flavors, and finally 1 out of the remaining 7 flavors. We can calculate this by multiplying the combinations together. So, for this scenario, the calculation would be: 9C2 x 8C2 x 7C1 = (9!)/(2!(9-2)!) x (8!)/(2!(8-2)!) x (7!)/(1!(7-1)!) = 36 x 28 x 7 = 8,568 ways of ordering.
To summarize:
1) All different: 15,120 ways of ordering.
2) Any combination: 126 ways of ordering.
3) Two scoops of one flavor, two scoops of the second flavor, and one of the third: 8,568 ways of ordering.
Remember, when calculating permutations and combinations, it is important to consider the principles of factorials and the formula for permutations and combinations.