Answer:
(a) 180
(b) 75
(c) 105
Explanation:
(a)
Restriction:
- 0 cannot be the hundreds
- each digit can be used only once
Hundreds possibility = 1, 2, 3, 4, 5, 6 → n = 6
Tens possibility = all 7 numbers except the hundreds → n = 7-1 = 6
Ones possibility = all 7 numbers except the hundreds & tens → n = 7-2 = 5
Total possibility = 6×6×5 = 180
(b)
Restriction:
- 0 cannot be the hundreds
- each digit can be used only once
- Odd numbers (ends with 1, 3 or 5)
[i] Tens ≠ 0
Ones possibility = 1, 3, 5 → n = 3
Tens possibility = 1, 2, 3, 4, 5, 6 except the ones → n = 6-1 = 5
Hundreds possibility = 1, 2, 3, 4, 5, 6 except the ones & tens → n = 6-2 = 4
Possibility = 3×5×4 = 60
[ii] Tens = 0
Ones possibility = 1, 3, 5 → n = 3
Tens possibility = 0 → n = 1
Hundreds possibility = 1, 2, 3, 4, 5, 6 except the ones → n = 6-1 = 5
Possibility = 3×1×5 = 15
∴ Total possibility = [i] + [ii]
= 60 + 15
= 75
(c)
Restriction:
- 0 cannot be the hundreds
- each digit can be used only once
- greater than 330 (not less than or equal to 330)
[i] numbers < 300
Hundreds = 1, 2 → n = 2
Tens = all 7 numbers except the hundreds → n = 7-1 = 6
Ones possibility = all 7 numbers except the hundreds & tens → n = 7-2 = 5
Possibility = 2×6×5 = 60
[ii] 300 ≤ numbers ≤ 330 AND Ones = 0, 1 or 2
Hundreds = 3 → n = 1
Tens = 0, 1, 2 → n = 3
Ones = 0, 1, 2 except the tens → n = 3-1 = 2
Possibility = 1×3×2 = 6
[iii] 300 ≤ numbers ≤ 330 AND Ones ≠ 0, 1 or 2
Hundreds = 3 → n = 1
Tens = 0, 1, 2 → n = 3
Ones = 4, 5, 6 → n = 3
Possibility = 1×3×3 = 9
∴ Total possibility = 180 - [i] - [ii] - [iii]
= 180 - 60 - 6 - 9
= 105