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Math

QuestionHow many ways can players choose 5 unique symbols from 9 total symbols for a password? Answer in simplest form: (95)\binom{9}{5}.

Studdy Solution

STEP 1

Assumptions1. There are9 different symbols available. . Each player chooses5 unique symbols for their password.
3. The order of the symbols in the password matters.

STEP 2

The problem is asking for the number of permutations of5 symbols chosen from9. The formula for permutations is given by(n,r)=n!(n−r)!(n, r) = \frac{n!}{(n-r)!}where- nn is the total number of options, - rr is the number of options chosen, - n!n! is the factorial of nn, and- (n−r)!(n-r)! is the factorial of (n−r)(n-r).

STEP 3

Now, plug in the given values for nn and rr into the permutations formula.
(9,5)=9!(9−5)!(9,5) = \frac{9!}{(9-5)!}

STEP 4

Calculate the factorial of 99 and (9−)(9-).
9!=9×8×7×6××4×3×2×19! =9 \times8 \times7 \times6 \times \times4 \times3 \times2 \times1(9−)!=4!=4×3×2×1(9-)! =4! =4 \times3 \times2 \times1

STEP 5

Substitute the calculated factorials into the permutations formula.
(9,5)=9!4!=9×8×7××5×4×3×2×14×3×2×1(9,5) = \frac{9!}{4!} = \frac{9 \times8 \times7 \times \times5 \times4 \times3 \times2 \times1}{4 \times3 \times2 \times1}

STEP 6

implify the fraction by cancelling out the common factors in the numerator and the denominator.
(9,5)=9×8××6×5(9,5) =9 \times8 \times \times6 \times5

STEP 7

Calculate the product to find the number of different ways the players can create a specific5 symbol password.
(9,5)=9××7×6×5=15,120(9,5) =9 \times \times7 \times6 \times5 =15,120There are15,120 different ways the players can create a specific5 symbol password.

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