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In a prediction contest, the outcomes of 10 football matches must each be guessed as win, draw, or loss. The number of possible entries that have at least 5 correct predictions is:

1. 3¹⁰ − ∑_(r=1)⁵ 10C_r 2^r
2. 3¹⁰ − ∑_(r=0)⁴ 10C_r 2^(10−r)
3. ∑_(r=1)⁵ 10C_r
4. ∑_(r=6)¹⁰ 10C_r 3^r

Correct answer: 3¹⁰ − ∑_(r=0)⁴ 10C_r 2^(10−r)

Solution

Each match has 1 correct and 2 wrong guesses, so exactly r correct = C(10,r)*2^(10-r). At least 5 correct = total 3^10 minus those with 0..4 correct = 3^10 - sum_(r=0)^4 C(10,r)2^(10-r).

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