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A passcode to enter a building is a sequence of 4 single digit numbers
(0-9), and repeated digits aren't allowed.
Suppose someone doesn't know the passcode and randomly guesses a sequence of 4 digits.
What is the probability that they guess the correct sequence?
Choose 1 answer:
(A)
(1)/(_(10)P_(4))
(B)
(1)/(_(10)C_(4))
(C)
(_(4)P_(4))/(_(10)P_(4))
(D)
((_(4)P_(2))*(_(4)P_(2)))/(_(10)P_(4))

A passcode to enter a building is a sequence of $4$ single digit numbers $(0-9)$ , and repeated digits aren't allowed. $\newline$ Suppose someone doesn't know the passcode and randomly guesses a sequence of $4$ digits. $\newline$ What is the probability that they guess the correct sequence? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{{ }_{10} \mathrm{P}_{4}}$ $\newline$ (B) $\frac{1}{{ }_{10} \mathrm{C}_{4}}$ $\newline$ (C) $\frac{{ }_{4} \mathrm{P}_{4}}{{ }_{10} \mathrm{P}_{4}}$ $\newline$ (D) $\frac{\left({ }_{4} \mathrm{P}_{2}\right) \cdot\left({ }_{4} \mathrm{P}_{2}\right)}{{ }_{10} \mathrm{P}_{4}}$

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Compare linear and exponential growth

Full solution

Q. A passcode to enter a building is a sequence of

4

single digit numbers

(0-9)

, and repeated digits aren't allowed.

\newline

Suppose someone doesn't know the passcode and randomly guesses a sequence of

4

digits.

\newline

What is the probability that they guess the correct sequence?

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{{ }_{10} \mathrm{P}_{4}}

\newline

(B)

\frac{1}{{ }_{10} \mathrm{C}_{4}}

\newline

(C)

\frac{{ }_{4} \mathrm{P}_{4}}{{ }_{10} \mathrm{P}_{4}}

\newline

(D)

\frac{\left({ }_{4} \mathrm{P}_{2}\right) \cdot\left({ }_{4} \mathrm{P}_{2}\right)}{{ }_{10} \mathrm{P}_{4}}

Understand the problem: Understand the problem. $\newline$ We need to calculate the probability of guessing a correct sequence of $4$ digits where no digit repeats and each digit can be from $0$ to $9$ .
Calculate total sequences: Calculate the total number of possible $4$ -digit sequences without repetition. $\newline$ Since the first digit can be any of the $10$ digits ( $0$ $-9$ ), the second digit can be any of the remaining $9$ digits, the third can be any of the remaining $8$ digits, and the fourth can be any of the remaining $7$ digits, we use the permutation formula for this. $\newline$ Total number of sequences = $10 \times 9 \times 8 \times 7 = 5040$ $\newline$ This is also denoted as $_{10}P_{4}$ , which is the number of permutations of $10$ items taken $4$ at a time.
Calculate probability: Calculate the probability of guessing the correct sequence. $\newline$ Since there is only one correct sequence, the probability of guessing it correctly is $1$ divided by the total number of possible sequences. $\newline$ Probability = $\frac{1}{_{10}P_{4}} = \frac{1}{5040}$
Match with options: Match the calculated probability with the given options. $\newline$ The calculated probability matches option (A), which is $\frac{1}{{}_{10}P_{4}}$ .

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