In how many different ways can the letters of the word GOODNESS be arranged? Option 1: 5040 Option 2: 20160 Option 3: 10080 Option 4: 10720

Answer: 10080. Solution: The word GOODNESS has 8 letters total. Writing out each letter: G - O - O - D - N - E - S - S Counting each letter: - G appears 1 time - O appears 2 times - D appears 1 time - N appears 1 time - E appears 1 time - S appears 2 times O repeats 2 times and S repeats 2 times. When letters repeat, the formula is: Number of arrangements = (n!)/(n_1! x n_2! x ...) where n is the total number of letters and n_1, n_2, ... represent how many times each letter repeats. Number of arrangements = (8!)/(2! x 2!) 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 8! = 40,320 2! = 2 Number of arrangements = (40,320)/(2 x 2) = (40,320)/4 = 10,080 Therefore, the letters of GOODNESS can be arranged in 10,080 different ways.

Quantitative Reasoning

Permutation & Combination

Medium

In how many different ways can the letters of the word GOODNESS be arranged?

5040

20160

10080

10720

✅ Correct Option: 3

The word GOODNESS has 8 letters total.

Writing out each letter: G - O - O - D - N - E - S - S

Counting each letter:

G appears 1 time
O appears 2 times
D appears 1 time
N appears 1 time
E appears 1 time
S appears 2 times

O repeats 2 times and S repeats 2 times.

When letters repeat, the formula is:

Number of arrangements $= \dfrac{n!}{n_1! \times n_2! \times ...}$

where $n$ is the total number of letters and $n_1, n_2, ...$ represent how many times each letter repeats.

Number of arrangements $= \dfrac{8!}{2! \times 2!}$

$8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

$8! = 40,320$

$2! = 2$

Number of arrangements $= \dfrac{40,320}{2 \times 2}$

$= \dfrac{40,320}{4}$

$= 10,080$

Therefore, the letters of GOODNESS can be arranged in 10,080 different ways.

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