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Maths-

General

Easy

Question

7 people leave their bags outside temple and returning after worshiping the deity picked one bag each at random. In how many ways at least one and at most three of them get their correct bags ?

⁷C₃ .9 + ⁷C₅ .44 + ⁷C₁ .265
⁷C₆ .265 + ⁷C₅ .9 + ⁷C₇ .44
⁷C₅ .9 + ⁷C₂ .44 + ⁷C₁ .265
None of these

The correct answer is: ⁷C₃ .9 + ⁷C₅ .44 + ⁷C₁ .265

No. of ways getting one correct
= ⁷C₁ 6! $open parentheses 1 minus fraction numerator 1 over denominator 1 factorial end fraction plus fraction numerator 1 over denominator 2 factorial end fraction minus fraction numerator 1 over denominator 3 factorial end fraction plus... plus fraction numerator 1 over denominator 6 factorial end fraction close parentheses$
= ⁷C₁.265
No. of ways getting two correct
= ⁷C₂. 5! $open parentheses 1 minus fraction numerator 1 over denominator 1 factorial end fraction plus fraction numerator 1 over denominator 2 factorial end fraction minus fraction numerator 1 over denominator 3 factorial end fraction plus... negative fraction numerator 1 over denominator 5 factorial end fraction close parentheses$
= ⁷C₂. 44
No. of ways getting three correct
= ⁷C₃.4! $open parentheses 1 minus fraction numerator 1 over denominator 1 factorial end fraction plus fraction numerator 1 over denominator 2 factorial end fraction minus fraction numerator 1 over denominator 3 factorial end fraction plus fraction numerator 1 over denominator 4 factorial end fraction close parentheses$
= ⁷C₃.9
Required no. of ways= ⁷C₃. 9 + ⁷C₂. 44 + ⁷C₁. 265

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