A security system generates 4-digit passcodes using distinct digits selected from the set . How many different 4-digit passcodes can be formed if the first digit must be odd and the last digit must be even?
Answer: 240
Answer
240
To form a valid passcode under the given constraints, we analyze each position step-by-step. The first digit has 4 possible choices (odd numbers: 1, 3, 5, 7), and the fourth digit has 3 possible choices (even numbers: 2, 4, 6). Since the problem specifies that the digits in the passcode must be distinct, choosing the first and last digits consumes 2 of the 7 available digits, leaving 5 digits for the second position and 4 digits for the third position. Applying the Fundamental Counting Principle gives a total of unique passcodes.
Step-by-Step Solution
Key Concept
Permutations with Position Restrictions