Question

1. engel has to make a 5-digit passcode from six different digits--1, 2, 3, 4, 5, and 6. no digit can be used more than once in a passcode.

a. how many combinations of five digits from the digits 1, 2, 3, 4, 5, and 6 are possible?

Explanation:

Step1: Identify the problem type

This is a combination problem where we need to choose 5 digits from 6 different digits. The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 6 \) (total digits) and \( k = 5 \) (digits to choose).

Step2: Substitute values into the formula

Substitute \( n = 6 \) and \( k = 5 \) into the combination formula:
\( C(6, 5)=\frac{6!}{5!(6 - 5)!}=\frac{6!}{5!1!} \)
Since \( n!=n\times(n - 1)\times\cdots\times1 \), we have \( 6! = 6\times5! \) and \( 1! = 1 \). So,
\( C(6, 5)=\frac{6\times5!}{5!\times1}=6 \)

Answer:

6