You own 19 CDs. You want to randomly arrange 9 of them in a CD rack. What is the probability that the rack ends up in alphabetical order?
The probability that the CDs are in alphabetical order is ?
To calculate the probability that the rack ends up in alphabetical order, we need to determine the total number of possible arrangements and the number of favorable arrangements (alphabetical order).
The total number of possible arrangements can be calculated using the formula for combinations, which is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items being selected. In this case, we have 19 CDs and we are selecting 9 of them, so the total number of possible arrangements is:
19C9 = 19! / (9!(19-9)!) = 19! / (9!10!) = (19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11) / (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
To determine the number of favorable arrangements (alphabetical order), we need to consider that there is only one specific order that qualifies as alphabetical. Therefore, the number of favorable arrangements is simply 1.
Therefore, the probability that the rack ends up in alphabetical order is:
1 / (19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 / (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1))
= 1 / 75582
So, the probability that the rack ends up in alphabetical order is approximately 0.0000132 (rounded to 7 decimal places) or 0.00132% (rounded to 2 decimal places).
To calculate the probability of the CDs being in alphabetical order, we need to start by determining the total number of possible arrangements of the 9 CDs in the rack.
Since there are 19 CDs in total, and we want to choose 9 of them, we can use the combination formula. The number of combinations, denoted as C(n, r), represents the number of ways to choose r items from a set of n items, without considering the order.
The formula for calculating combinations is:
C(n, r) = n! / (r!(n - r)!)
In this case, we have:
n = 19 (total number of CDs)
r = 9 (number of CDs to arrange in the rack)
Using the combination formula, we can calculate the total number of possible arrangements:
C(19, 9) = 19! / (9!(19 - 9)!)
Simplifying this expression:
C(19, 9) = 19! / (9! * 10!)
Next, we need to determine the number of arrangements where the CDs are in alphabetical order. Since we want the CDs to be in alphabetical order, the first CD chosen has to be the earliest in the alphabet, the second CD has to be the next earliest, and so on. This means there is only one possible arrangement that satisfies this condition.
Now we can calculate the probability of the CDs being in alphabetical order by dividing the number of arrangements where the CDs are in alphabetical order by the total number of possible arrangements.
Probability = Number of arrangements with CDs in alphabetical order / Total number of possible arrangements
In this case, the probability would be:
Probability = 1 / C(19, 9)
You would need to calculate C(19, 9) and then calculate the reciprocal of that value to get the probability.
18.8
we must assume that each CD's name begins with a different letter
number of ways to arrange 9 of 19 items
= P(19,9)
= 19x18x17x16x15x14x13x12x11
= 3.3522... x 10^10
so only one of these is in the correct order
prob(correct order) = 1/3.3522 x 10^10
= appr 2.98 x 10^-11