Numerical answer
In how many distinct arrangements of the letters of the word BANANA are the two N's not next to each other?
Show answer and solution
Answer: 40
Gap method: first arrange the non-N letters B, A, A, A — 4!/3! = 4 ways.
They create 5 gaps (_ B _ A _ A _ A _). Place the two identical N's in 2 different gaps: C(5, 2) = 10 ways.
Total = 4 × 10 = 40.
Check by complement: all arrangements = 6!/(3! 2!) = 60; with NN glued together: 5!/3! = 20; so 60 − 20 = 40 ✓.
They create 5 gaps (_ B _ A _ A _ A _). Place the two identical N's in 2 different gaps: C(5, 2) = 10 ways.
Total = 4 × 10 = 40.
Check by complement: all arrangements = 6!/(3! 2!) = 60; with NN glued together: 5!/3! = 20; so 60 − 20 = 40 ✓.
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