PnC Basics
Find the number of all positive integers up to 500 with non-repeating digits.
| Type of Number | Digit Positions | Choices at Each Step | Reasoning | Count |
|---|---|---|---|---|
| 1-digit | _ | 1-9 → 9 choices | Any digit 1-9 is valid | 9 |
| 2-digit | Tens digit | 1-9 → 9 choices | Tens digit can’t be 0 | 81 |
| Units digit | 0-9 except tens digit → 9 choices | Can’t repeat tens digit | ||
| 3-digit (100-499) | Hundreds digit | 1-4 → 4 choices | First digit ≤ 4 so number ≤ 499 | 288 |
| Tens digit | 0-9 except hundreds digit → 9 choices | Must differ from hundreds digit | ||
| Units digit | 0-9 except hundreds & tens digit → 8 choices | Must differ from both |
Step 2: 2-digit numbers (10–99): Tens digit 1-9 → 9 choices, units digit ≠ tens → 9 choices → 9×9 = 81
Step 3: 3-digit numbers (100–499): Hundreds 1-4 → 4, tens ≠ hundreds → 9, units ≠ previous → 8 → 4×9×8 = 288
Total = 9 + 81 + 288 = 378
Indices + Logs
If 3^a = 4, 4^b = 5, 5^c = 6, 6^d = 7, 7^e = 8, and 8^f = 9, then what is the value of abcdef?
abcdef = log9 / log3 = log base 3 of 9 = 2
Functions + Quadratic
For any non-zero real x, let f(x) + 2 f(1/x) = 3x. Find the sum of all possible values of x for which f(x) = 3.
f(1/x) + 2 f(x) = 3 / x
Multiply 2nd by 2 and subtract:
3 f(x) = (6/x) – 3x → f(x) = (2/x) – x
Set f(x)=3: (2/x)-x=3 → 2 – x² = 3x → x² + 3x -2 =0
x = (-3 ± √17)/2
Sum = -3
Average Problem
The average of three distinct real numbers is 28. If the smallest number is increased by 7 and the largest number is reduced by 10, the order of the numbers remains unchanged, and the new arithmetic mean becomes 2 more than the middle number, while the difference between the largest and the smallest numbers becomes 64. Then, the largest number in the original set of three numbers is _____
(c-10)-(a+7)=64 → c-a=81
Add: 2c=140 → c=70
Answer: 70
Gopi and Ravi Profit
Gopi marks a price on a product in order to make 20% profit. Ravi gets 10% discount on this marked price, and thus saves Rs 15. Then, the profit, in rupees, made by Gopi by selling the product to Ravi is:
SP=135, CP=112.5
Profit = 22.5
System of Equations
For some constant real numbers p, k and a, consider the system of linear equations: px – 4y = 2 and 3x + ky = a. A necessary condition for the system to have no solution for (x,y) is:
Not coincident: 2a + k ≠ 0
Milk and Water
A 300 litre container. Water x litres, rest milk. Remove 2x litres solution, refill water. Final solution 72% milk. Find x.
x=30
Rajesh & Vimal Land
Rajesh owns 20 hectares, Vimal owns 30 hectares with wheat and mustard in 5:3 ratio. Total wheat:mustard is 11:9. Find Rajesh’s ratio.
Aman Bank Investment
Aman invests Rs 4000 in a bank at a certain rate of interest, compounded annually. If the ratio of the value of the investment after 3 years to the value after 5 years is 25:36, then the minimum number of years required for the value to exceed Rs 20000 is _____
4000*(1.2)^n > 20000 → (1.2)^n>5
n=8:4.3, n=9:5.16 → n=9
