Logical Arrangement Practice Questions Practice Test Worksheets
Logical arrangement questions test systematic counting. You may need to count code patterns, outfit choices, handshakes, digit arrangements, or letter shuffles without missing cases or counting the same case twice.
For students who can count simple choices, but get stuck when restrictions, pairings, and repeated arrangements change the total.
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What Do Logical Arrangement Questions Look Like?
Logical arrangement questions ask you to calculate the number of possible outcomes, permutations, or combinations given a set of items and restrictions. The key is applying the multiplication principle and accounting for ordering rules. Here is a worked example.
Product Rule Choices
Question: Sean has 5 different hats and 3 different scarves. Each outfit uses one hat and one scarf. If Sean must choose one particular scarf, how many outfits are possible?
Worked Method
Step 1: The scarf choice is fixed, so it contributes only 1 option.
Step 2: There are still 5 possible hats.
1 × 5 = 5 outfits.
Tip: A fixed choice does not disappear. It counts as 1 option, not 0.
Rules to Help You Solve Logical Arrangement Questions
To solve logical arrangement questions like the example above, break complex scenarios into independent stage choices. Follow these four rules:
🔢 Rule 1: Count choices, then multiply
Check how many choices each position has, allowing for any restrictions, then multiply. This works when each earlier choice leaves the same number of options for the next position.
Example: Choose one of 4 letters for each of the first two positions, then one of 4 digits. Letters may repeat, so there are 4 × 4 × 4 = 64 codes.
🚫 Rule 2: Restrictions reduce choices
If letters or digits cannot repeat, or a fixed symbol must be used, reduce the choices before multiplying.
Example: Two different letters from 4 choices gives 4 × 3 = 12 ordered pairs.
🤝 Rule 3: Avoid counting pairs twice
For a handshake or a single match between two people, A with B is the same as B with A. When you count each person with every other person, each pair appears twice. Divide that total by 2.
Example: 6 people give 6 × 5 ÷ 2 = 15 pairs.
🔤 Rule 4: Arrangements lose one choice each time
When arranging different letters or digits with no repeats and no extra restrictions, each position has one fewer choice than the last. Check for extra rules first: for example, a number with two or more digits cannot start with zero.
Example: CHAIR has 5 different letters, so 5 × 4 × 3 × 2 × 1 = 120 arrangements. These include letter orders that are not real words.
Two More Worked Examples
Here are two more examples demonstrating restricted pattern arrangements and avoiding unordered pairing traps.
Restricted Code Patterns
Question: A ticket code uses two letters chosen from I, L, O, S, then one number chosen from 1, 2, 3, 4. Repeated letters are allowed. How many codes have two different letters and end in an even number?
Worked Method
Step 1: There are 4 choices for the first letter.
Step 2: The second letter must be different, so only 3 choices remain.
Step 3: The even numbers are 2 and 4, so there are 2 number choices.
4 × 3 × 2 = 24 codes.
Tip: Repeats are allowed in the full code system, but this particular question adds the restriction "two different letters".
Pairings Are Not Ordered
Question: 6 debaters are paired with every other debater once. How many debates are there in total?
Worked Method
Step 1: Each of the 6 people could be paired with 5 others, giving 6 × 5 = 30 pair counts.
Step 2: Each debate has been counted twice. For example, A with B and B with A are the same debate.
30 ÷ 2 = 15 debates.
Tip: Divide by 2 when your count includes each pair in both orders. Do not divide for codes, where AB and BA are different codes.
