STRESSED word

Each letter in STRESSED is printed on identical cards, one letter per card, and assembled in random order. Calculate the probability that the cards spell DESSERTS when assembled.

Correct result:

p =  0.0298 %

Solution:

$$\begin{gathered} n_1=1 \\ n=8 \\ n(S)=3 \\ n(T)=1 \\ n(R)=1 \\ n(E)=2 \\ n(D)=1 \\ {n}_2={\displaystyle \frac{n!}{n(S)!\cdot n(T)!\cdot n(R)!\cdot n(E)!\cdot n(D)!}} \\ {n}_2={\displaystyle \frac{8!}{3!\cdot 2!}}=3360 \\ p=100\cdot {\displaystyle \frac{n_1}{n_2}}=100\cdot {\displaystyle \frac{1}{3360}}={\displaystyle \frac{5}{168}}=0.0298\mathrm{\%} \end{gathered}$$

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Showing 6 comments:

Help

where does 3! and 2! come from?

1 year ago  Like

Dr Math

some characters are repeated: 3xS + 2xE; so there is no difference between first second and third S or 1st or 2nd E char.

1 year ago  Like

Math student

How did you get 3360

1 year ago  Like

Math student

dividing the factorials of 8!= 40,320 / 3!= 6 x 2!=2 so 40,320/12=3360
8!= 8x7x6x5x4x3x2x1= 40,320

1 year ago  Like

Math student

why is n1 = 1?

11 months ago  Like

Matematik

n1 is the count of positive card (word DESSERTS) -> only one. n2 are the count of all combinations

11 months ago  Like

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