A security system requires a 4-digit passcode created using the digits 1, 2, 3, 4, 5, 6, a… — ACT

A security system requires a 4-digit passcode created using the digits $1, 2, 3, 4, 5, 6,$ and $7$ . No digit may be repeated in a passcode. If the first digit and the last digit must both be odd numbers, how many different passcodes can be formed?

Answer: 240 passcodes

Answer

240 passcodes can be formed under the given conditions.

To form a 4-digit passcode with distinct digits where the first and last digits are odd: first choose the first digit from the 4 available odd digits (

1, 3, 5, 7

). Next, choose the fourth digit from the remaining 3 odd digits. Then, choose the second digit from the remaining 5 available digits in the full set, and the third digit from the remaining 4 available digits. Multiplying these choices gives

4 \times 5 \times 4 \times 3 = 240

Step-by-Step Solution

Categorize available digits into odd and even sets

4 odd digits ({1, 3, 5, 7}) and 3 even digits ({2, 4, 6}) out of 7 total digits

The first and last positions require odd digits, so we need to track odd digits separately.

Determine options for the first and last positions

4 options for position 1; 3 options for position 4

Position 1 must be odd (4 choices). Because digits cannot repeat, position 4 has 3 remaining odd choices.

Determine options for the second and third positions

5 options for position 2; 4 options for position 3

Two digits have already been assigned. Out of 7 total digits, 5 remain for position 2, leaving 4 for position 3.

Multiply choices across all positions

4 * 5 * 4 * 3 = 240

By the Fundamental Counting Principle, the total number of combinations is the product of the number of choices at each step.

Key Concept

Fundamental Counting Principle with Position Restrictions and Non-Replacement

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