How many six-letter sequences are possible that use the letters c, i, c, c, i, y?
notice we got 3 c's and 2 i's
suppose the c's and i's are all different, therefore distinguishable.
(pretend they are all of a different colour)
then they can be arranged in 6! ways.
but we CAN'T tell the c's apart, so we have counted 3! of them that look alike.
similarly we counted 2! of them with i's that look alike,
these have to be divided out.
so the number of ways is 6!/(3!2!) = 60
this is the general procedure for these types of questions.
first take the factorial of the total, then divide by the factorials of each duplication count
e.g. aaabbcccc
can be arranged in 9!/(3!2!4!)ways.
To find the number of six-letter sequences that can be formed using the letters c, i, c, c, i, y, we can use the concept of permutations.
First, let's count the number of times each letter appears:
- The letter 'c' appears 3 times.
- The letter 'i' appears 2 times.
- The letter 'y' appears 1 time.
Now, let's calculate the total number of permutations:
We have a total of 6 positions to fill with the given letters.
For the first position, we have 6 options (c, i, c, c, i, y).
For the second position, we have 5 options remaining (after using one of the letters from the first position).
For the third position, we have 4 options remaining (after using one of the letters from the previous positions).
For the fourth position, we have 3 options remaining (after using one of the letters from the previous positions).
For the fifth position, we have 2 options remaining (after using one of the letters from the previous positions).
Finally, for the sixth position, we have 1 option remaining (after using all the letters from the previous positions).
To calculate the total number of permutations, we multiply these numbers together:
6 * 5 * 4 * 3 * 2 * 1 = 720
Therefore, there are 720 possible six-letter sequences that can be formed using the letters c, i, c, c, i, y.
To find the number of six-letter sequences that can be formed using the letters c, i, c, c, i, y, we can use the concept of permutations.
A permutation is an arrangement of objects in a specific order. In this case, we have six objects (letters) to arrange, but since some of the objects are repeated (the letter 'c' appears twice and the letter 'i' appears twice), we need to take into account the repetition.
To approach this problem, we need to consider two scenarios:
1. If all the letters were distinct: If all the letters were distinct, we would have six distinct objects to arrange, which can be done in 6! (6 factorial) ways. The factorial of a number represents the product of all positive integers less than or equal to that number. So, 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720.
2. Taking into account the repetitions: We need to adjust for the repeated letters 'c' and 'i'. Since there are two 'c's and two 'i's, we need to divide the total number of arrangements from scenario 1 by the number of arrangements of the repeated letters. The number of arrangements of 'c' can be found using 2! (2 factorial) and the number of arrangements of 'i' can also be found using 2!.
So, the total number of arrangements is 6! / (2! x 2!) = 720 / (2 x 2) = 180.
Therefore, there are 180 possible six-letter sequences that can be formed using the letters c, i, c, c, i, y.