Permutations & Combinations
Exercise 6.1 • Fundamental Principle of Counting
1. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) repetition is allowed? (ii) repetition is not allowed?
Given Digits
$\{1, 2, 3, 4, 5\}$ (Total 5 digits)
(i) Repetition Allowed
Positions
Hundreds $\times$ Tens $\times$ Units
Ways
$5 \times 5 \times 5$
Total Numbers = 125
(ii) Repetition Not Allowed
Ways
$5 \times 4 \times 3$
Total Numbers = 60
2. How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
Digits
$\{1, 2, 3, 4, 5, 6\}$ (Total 6 digits)
Constraint
Even numbers must end in 2, 4, or 6.
Unit Place
3 ways (2, 4, or 6)
Ten’s Place
6 ways (Repetition allowed)
Hun’s Place
6 ways (Repetition allowed)
Calculation
$6 \times 6 \times 3$
Total Even Numbers = 108
3. How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
Letters
First 10 alphabets ($n=10$)
Code Length
4 letters (4 places to fill)
1st Place
10 ways
2nd Place
9 ways (No repetition)
3rd Place
8 ways
4th Place
7 ways
Calculation
$10 \times 9 \times 8 \times 7$
Total Codes = 5040
4. How many 5-digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
Digits
$\{0, 1, …, 9\}$ (Total 10 digits)
Structure
$\boxed{6}\boxed{7}\boxed{?}\boxed{?}\boxed{?}$
Fixed
First 2 places are fixed (1 way each). Digits 6 and 7 are used.
Remaining
8 digits available for 3 places.
Calculation
$1 \times 1 \times 8 \times 7 \times 6$
Total Telephone Numbers = 336
5. A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?
Event
Tossing a coin has 2 outcomes (Head, Tail).
1st Toss
2 outcomes
2nd Toss
2 outcomes
3rd Toss
2 outcomes
Total
$2 \times 2 \times 2$
Total Outcomes = 8
6. Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other?
Available
5 different colored flags.
Requirement
2 flags arranged vertically.
Upper Flag
5 ways (any of the 5 colors).
Lower Flag
4 ways (cannot repeat the upper color).
Calculation
$5 \times 4$
Total Signals = 20