How many 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6 and 7, if no digit is repeated in a number?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 840
The four places are filled one after another, and since no digit may repeat, each place has one choice fewer than the place before. There is no 0, so any digit can stand first.
Thousands place: 7 choices
Hundreds place: 6 choices left
Tens place: 5 choices left
Units place: 4 choices left
Numbers \(=7\times6\times5\times4={}^7P_4=840\)
Why the other options are incorrect
Option (1) 2401 — Incorrect. This is \(7^4\), the count when digits may repeat.
Option (2) 35 — Incorrect. This is \(^7C_4\); it only picks 4 digits and does not arrange them into numbers.
Option (4) 5040 — Incorrect. This is \(7!\), which uses all 7 digits and so gives 7-digit numbers.