A lock has a code of $5$ numbers from $1$ to $10$ . If no numbers in the code are allowed to repeat, how many different codes could be made? $\newline$ Answer:

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Q. A lock has a code of

5

numbers from

1

10

. If no numbers in the code are allowed to repeat, how many different codes could be made?

\newline

Answer:

Determine first digit choices: Determine the number of choices for the first digit of the code. $\newline$ Since there are $10$ numbers to choose from and no restrictions for the first digit, there are $10$ possible choices for the first number in the code.
Determine second digit choices: Determine the number of choices for the second digit of the code. $\newline$ After choosing the first digit, there are $9$ remaining numbers to choose from for the second digit, since no number can be repeated.
Determine third digit choices: Determine the number of choices for the third digit of the code. $\newline$ Similarly, after choosing the first two digits, there are $8$ remaining numbers to choose from for the third digit.
Determine fourth digit choices: Determine the number of choices for the fourth digit of the code. $\newline$ After the first three digits have been chosen, there are $7$ remaining numbers to choose from for the fourth digit.
Determine fifth digit choices: Determine the number of choices for the fifth and final digit of the code. $\newline$ After choosing the first four digits, there are $6$ remaining numbers to choose from for the fifth digit.
Calculate total number of codes: Calculate the total number of different codes that can be made. $\newline$ To find the total number of different codes, multiply the number of choices for each digit together. $\newline$ Total number of codes = $10 \times 9 \times 8 \times 7 \times 6$
Perform final multiplication: Perform the multiplication to find the final answer. $\newline$ Total number of codes = $10 \times 9 \times 8 \times 7 \times 6 = 30240$