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 ( $7!$ divided by 212!) different ways to arrange the letters in DOTPLOT. There would be $7!$ arrangements if they were all different, reduced by a factor of $2!$ 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? $\square$

Studdy Solution

STEP 1

1. The word "OHIO" consists of 4 letters.
2. The letter 'O' appears twice in the word "OHIO".

STEP 2

1. Calculate the total permutations if all letters were unique.
2. Adjust for the repeated letter.

STEP 3

Calculate the total number of permutations if all letters in "OHIO" were unique. Since there are 4 letters, the total permutations would be:
$4!$

STEP 4

Since the letter 'O' is repeated, we need to adjust for this repetition. We divide by the factorial of the number of times 'O' appears, which is $2!$ . Therefore, the number of distinct permutations is:
$\frac{4!}{2!}$

STEP 5

Calculate the factorials:
$4! = 4 \times 3 \times 2 \times 1 = 24$ $2! = 2 \times 1 = 2$

STEP 6

Divide the total permutations by the factorial of the repetitions:
$\frac{24}{2} = 12$
The number of distinct ways to arrange the letters in the word "OHIO" is:
$\boxed{12}$

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Studdy Solution

STEP 1

1. The word "OHIO" consists of 4 letters.
2. The letter 'O' appears twice in the word "OHIO".

STEP 2

1. Calculate the total permutations if all letters were unique.
2. Adjust for the repeated letter.

STEP 3

Calculate the total number of permutations if all letters in "OHIO" were unique. Since there are 4 letters, the total permutations would be:
$4!$

STEP 4

Since the letter 'O' is repeated, we need to adjust for this repetition. We divide by the factorial of the number of times 'O' appears, which is $2!$ . Therefore, the number of distinct permutations is:
$\frac{4!}{2!}$

STEP 5

Calculate the factorials:
$4! = 4 \times 3 \times 2 \times 1 = 24$ $2! = 2 \times 1 = 2$

STEP 6

Divide the total permutations by the factorial of the repetitions:
$\frac{24}{2} = 12$
The number of distinct ways to arrange the letters in the word "OHIO" is:
$\boxed{12}$

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Studdy Solution

STEP 1

1. The word "OHIO" consists of 4 letters.2. The letter 'O' appears twice in the word "OHIO".

STEP 2

1. Calculate the total permutations if all letters were unique.2. Adjust for the repeated letter.

STEP 3

Calculate the total number of permutations if all letters in "OHIO" were unique. Since there are 4 letters, the total permutations would be:4! 4! 4!

STEP 4

Since the letter 'O' is repeated, we need to adjust for this repetition. We divide by the factorial of the number of times 'O' appears, which is 2! 2! 2!. Therefore, the number of distinct permutations is:4!2! \frac{4!}{2!} 2!4!​

STEP 5

Calculate the factorials:4!=4×3×2×1=24 4! = 4 \times 3 \times 2 \times 1 = 24 4!=4×3×2×1=24 2!=2×1=2 2! = 2 \times 1 = 2 2!=2×1=2

STEP 6

Divide the total permutations by the factorial of the repetitions:242=12 \frac{24}{2} = 12 224​=12 The number of distinct ways to arrange the letters in the word "OHIO" is:12 \boxed{12} 12​

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1. The word "OHIO" consists of 4 letters.
2. The letter 'O' appears twice in the word "OHIO".

1. Calculate the total permutations if all letters were unique.
2. Adjust for the repeated letter.

Calculate the total number of permutations if all letters in "OHIO" were unique. Since there are 4 letters, the total permutations would be:
$4!$

Since the letter 'O' is repeated, we need to adjust for this repetition. We divide by the factorial of the number of times 'O' appears, which is $2!$ . Therefore, the number of distinct permutations is:
$\frac{4!}{2!}$

Calculate the factorials:
$4! = 4 \times 3 \times 2 \times 1 = 24$ $2! = 2 \times 1 = 2$

Divide the total permutations by the factorial of the repetitions:
$\frac{24}{2} = 12$
The number of distinct ways to arrange the letters in the word "OHIO" is:
$\boxed{12}$