Find the probability that if the letters of the word "parallel" are randomly arranged that the L's will not be together.
In class, I'm studying permutations and combinations. The solutions stated the no. of permutations where 3 L's are together is 6!/2!. Could you please explain why?
Thanks in advance
Treat the LLL as if they were one element, say X
so we have
X a a p r e, 6 elements of which two are alike, the two a's
number of ways to arrange them is 6!/2!
(remember we divide by 2! because of the two "alikes" )
So in the first part, the number of ways to arrange the original is
8!/(3!2!) = 3360
The number of ways the LLL is together is 6!/2! = 360
Number of ways the L's are NOT together = 3360 - 360 = 3000
To find the probability that the L's will not be together when the letters of the word "parallel" are randomly arranged, we need to calculate the number of favorable outcomes (arrangements where the L's are not together) and divide it by the total number of possible outcomes (all the permutations of the word).
Let's first determine the total number of possible outcomes. The word "parallel" has 8 letters, but the two "L's" are repeated. So, the total number of possible permutations is given by
8! / (2! x 2!) = 8! / (2!^2),
where "8!" represents the factorial of 8, which is the product of all positive integers from 1 to 8.
Now, let's calculate the number of favorable outcomes, where the "L's" are not together. To accomplish this, we can treat the three "L's" as a group (LLL). With this, we only need to consider the arrangement of 6 elements (LLLL in a group and the two "A's" and one "R" as separate entities):
6! / 2!.
The 6! represents the number of permutations of the 6 elements, and the 2! takes into account the repeats of the "A's." So, we divide by 2! because "A" is repeated twice.
Therefore, the probability that the "L's" will not be together is given by:
(6! / 2!) / (8! / (2! x 2!)).
Simplifying, we have 6! / (8! / (2! x 2!)) = (6! x (2! x 2!)) / 8!.
To calculate the exact value of this probability, you can input this expression into a calculator or use the factorial function to evaluate it.
To find the probability that the L's will not be together when the letters of the word "parallel" are randomly arranged, we need to first find the total number of possible arrangements of the word "parallel."
The word "parallel" has 8 letters, including 3 L's. We can arrange these 8 letters in 8! (factorial) ways. However, since the letters are not all distinct (the L's are repeated), we need to divide by the factorials of the number of times each repeated letter appears.
In this case, the letter L appears 3 times. So, we divide the total number of arrangements by 3! (3 factorial) to account for the repeated L's.
Therefore, the total number of possible arrangements of the word "parallel" is given by:
Total number of arrangements = 8! / (3!)
Now, to find the number of arrangements where the L's are together, we can treat the 3 L's as a single unit. So, we have 6 units to arrange, which include the single L, the other single L, the group of 3 L's, and the remaining 3 distinct letters (P, A, R).
The number of arrangements of these 6 units, taking into account the repeated letters, can be found by dividing 6! (6 factorial) by the factorials of the number of times each repeated letter appears (2! for the repeated L's).
Therefore, the number of arrangements where the 3 L's are together is given by:
Number of arrangements with 3 L's together = 6! / (2!)
Now, to find the number of arrangements where the L's are not together, we subtract the number of arrangements where the 3 L's are together from the total number of arrangements.
Number of arrangements with L's not together = Total number of arrangements - Number of arrangements with 3 L's together
Number of arrangements with L's not together = (8! / (3!)) - (6! / (2!))
Finally, to find the probability, we divide the number of arrangements with L's not together by the total number of arrangements:
Probability = (8! / (3!)) - (6! / (2!)) / (8! / (3!))
Simplifying the expression, you can calculate the numerical value using a calculator or software to find the probability.