Math / Discrete

QuestionHow many ways can the letters in MAXIMUM be arranged? Well, there are 7 letters, so that would be $7!$ if they were all different. But that counts MAXIMUM as a new arrangement (switching the M's around). To avoid recounting these "arrangements," note that every ordering appears $3!=6$ times because of the matching M's. To eliminate all those duplicates from consideration, divide by 3!. This reveals that there are actually 840 (7! divided by 3!) distinct permutations of the letters in MAXIMUM. Similarly, there would be 1260 (71 divided by 2121) different ways to arrange the letters in DOTPLOT. There would be $7!$ arrangements if they were all different, reduced by a factor of 21 for the matching O's and $2!$ for the matching Ts. Complete parts a through f below. a) How many ways can the letters in the word OHIO be arranged?
12 b) How many ways can the letters in the word ALASKA be arranged?
120 c) How many ways can the letters in the word MONTANA be arranged?
1260 d) How many ways can the letters in the word ARKANSAS be arranged? $\square$

Studdy Solution

STEP 1

1. We are arranging the letters of the word "ARKANSAS".
2. The word "ARKANSAS" consists of 8 letters.
3. The letter 'A' appears 3 times, and the letter 'S' appears 2 times.

STEP 2

1. Calculate the total number of permutations if all letters were distinct.
2. Adjust for repeated letters by dividing by the factorials of the counts of repeated letters.

STEP 3

Calculate the total number of permutations if all letters were distinct. Since there are 8 letters in "ARKANSAS", we have:
$8!$

STEP 4

Adjust for repeated letters. The letter 'A' appears 3 times, and the letter 'S' appears 2 times. We divide by the factorial of these counts:
$\frac{8!}{3! \times 2!}$

STEP 5

Calculate the factorials:
$8! = 40320$ $3! = 6$ $2! = 2$

STEP 6

Substitute the factorial values into the equation:
$\frac{40320}{6 \times 2}$ $\frac{40320}{12}$

STEP 7

Perform the division to find the number of distinct permutations:
$\frac{40320}{12} = 3360$
The number of ways the letters in the word "ARKANSAS" can be arranged is:
$\boxed{3360}$

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Studdy Solution

STEP 1

1. We are arranging the letters of the word "ARKANSAS".
2. The word "ARKANSAS" consists of 8 letters.
3. The letter 'A' appears 3 times, and the letter 'S' appears 2 times.

STEP 2

1. Calculate the total number of permutations if all letters were distinct.
2. Adjust for repeated letters by dividing by the factorials of the counts of repeated letters.

STEP 3

Calculate the total number of permutations if all letters were distinct. Since there are 8 letters in "ARKANSAS", we have:
$8!$

STEP 4

Adjust for repeated letters. The letter 'A' appears 3 times, and the letter 'S' appears 2 times. We divide by the factorial of these counts:
$\frac{8!}{3! \times 2!}$

STEP 5

Calculate the factorials:
$8! = 40320$ $3! = 6$ $2! = 2$

STEP 6

Substitute the factorial values into the equation:
$\frac{40320}{6 \times 2}$ $\frac{40320}{12}$

STEP 7

Perform the division to find the number of distinct permutations:
$\frac{40320}{12} = 3360$
The number of ways the letters in the word "ARKANSAS" can be arranged is:
$\boxed{3360}$

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Studdy Solution

STEP 1

1. We are arranging the letters of the word "ARKANSAS".2. The word "ARKANSAS" consists of 8 letters.3. The letter 'A' appears 3 times, and the letter 'S' appears 2 times.

STEP 2

1. Calculate the total number of permutations if all letters were distinct.2. Adjust for repeated letters by dividing by the factorials of the counts of repeated letters.

STEP 3

Calculate the total number of permutations if all letters were distinct. Since there are 8 letters in "ARKANSAS", we have:8! 8! 8!

STEP 4

Adjust for repeated letters. The letter 'A' appears 3 times, and the letter 'S' appears 2 times. We divide by the factorial of these counts:8!3!×2! \frac{8!}{3! \times 2!} 3!×2!8!​

STEP 5

Calculate the factorials:8!=40320 8! = 40320 8!=40320 3!=6 3! = 6 3!=6 2!=2 2! = 2 2!=2

STEP 6

Substitute the factorial values into the equation:403206×2 \frac{40320}{6 \times 2} 6×240320​ 4032012 \frac{40320}{12} 1240320​

STEP 7

Perform the division to find the number of distinct permutations:4032012=3360 \frac{40320}{12} = 3360 1240320​=3360 The number of ways the letters in the word "ARKANSAS" can be arranged is:3360 \boxed{3360} 3360​

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1. We are arranging the letters of the word "ARKANSAS".
2. The word "ARKANSAS" consists of 8 letters.
3. The letter 'A' appears 3 times, and the letter 'S' appears 2 times.

1. Calculate the total number of permutations if all letters were distinct.
2. Adjust for repeated letters by dividing by the factorials of the counts of repeated letters.

Calculate the total number of permutations if all letters were distinct. Since there are 8 letters in "ARKANSAS", we have:
$8!$

Adjust for repeated letters. The letter 'A' appears 3 times, and the letter 'S' appears 2 times. We divide by the factorial of these counts:
$\frac{8!}{3! \times 2!}$

Calculate the factorials:
$8! = 40320$ $3! = 6$ $2! = 2$

Substitute the factorial values into the equation:
$\frac{40320}{6 \times 2}$ $\frac{40320}{12}$

Perform the division to find the number of distinct permutations:
$\frac{40320}{12} = 3360$
The number of ways the letters in the word "ARKANSAS" can be arranged is:
$\boxed{3360}$