Using the letters in ADJUSTING without repeats,
How many different nine letter arrangements could be made?
How many of these nine-letter arrangements have a consonant in fifth position and J in sixth position?
What special name is given to the sixth position in the last question>
looks like all the letters are different
1) 9!
2) put those letters in the required spots, leaving 7 remaining to arrange
so 7!
3) ???
To find the number of different nine-letter arrangements that can be made using the letters in ADJUSTING without repeats, we can use the concept of permutations.
1. Number of different nine-letter arrangements:
Since there are nine letters in the word ADJUSTING, we have 9 slots to fill. We have 9 choices for the first slot, 8 choices for the second slot (since one letter has already been used), 7 choices for the third slot, and so on. Thus, the number of different arrangements is calculated as:
9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880
Therefore, there are 362,880 different nine-letter arrangements that can be made using the letters in ADJUSTING without repeats.
2. Number of nine-letter arrangements with a consonant in the fifth position and J in the sixth position:
Since we need a consonant in the fifth position, we have to choose a consonant from ADSTNG, which leaves us with 6 choices. Similarly, for the sixth position, we have to choose the letter J, so we have only one choice. The remaining letters can be arranged in the remaining 7 slots, which is the same as calculating the number of different seven-letter arrangements using the letters A, D, U, S, T, I, and N.
To calculate the number of seven-letter arrangements, we have 7 choices for the first slot, 6 choices for the second slot, 5 choices for the third slot, and so on. Thus, the number of different arrangements is calculated as:
7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040
Therefore, there are 5,040 different nine-letter arrangements with a consonant in the fifth position and the letter J in the sixth position.
3. Special name for the sixth position in the last question:
The special name given to the sixth position in the last question is the "sixth position." There is no specific special name for this position other than its numerical order.