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Flip to reveal answersHow do you count shortest routes across a grid (right/up only)?
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All 8 Flashcards — Grid arrangements
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Question
How do you count shortest routes across a grid (right/up only)?
Answer
A route is an arrangement of right and up moves. Choose which moves go 'up' (or right) with ⁿCᵣ.
Question
How many shortest routes cross an a-wide, b-tall grid?
Answer
Make a right moves and b up moves (a + b total); choose the b up moves: ⁽ᵃ⁺ᵇ⁾Cᵦ.
Question
How do you count arrangements of a word with repeated letters?
Answer
Divide n! by the factorial of each repeated letter: n! ÷ (p! q! …).
Question
Why divide by the repeats' factorials?
Answer
Swapping two identical letters gives the same word, so plain n! counts each arrangement several times; dividing removes the duplicates.
Question
Arrangements of BANANA?
Answer
6 letters with A×3, N×2: 6! ÷ (3! 2!) = 720 ÷ 12 = 60.
Question
Shortest routes across a 4-wide, 3-tall grid?
Answer
7 moves, choose 3 up: ⁷C₃ = 35.
Question
Grid routes vs word arrangements — what's the link?
Answer
A grid route is a word made of two letters (R and U), so both use the same arrangement idea.
Question
Arrangements of MISSISSIPPI?
Answer
11 letters with S×4, I×4, P×2: 11! ÷ (4! 4! 2!) = 34650.
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Full study notes for Grid arrangements
Topic 1.10 hub
Counting & binomial (HL only)
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