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Solutions from a student (dragged) 19

# Solutions from a student (dragged) 19 - k-1 committee...

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Problem 10 (committee’s with a chair) Part (a): We can select a committee with k members in n k ways. Selecting a chairper- son from the k committee members gives k n k possible choices. Part (b): If we choose the non chairperson members first this can be done in n k - 1 ways. We then choose the chairperson based on the remaining n - k + 1 people. Combining these two we have ( n - k + 1) n k - 1 possible choices. Part (c): We can first pick the chair of our committee in
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Unformatted text preview: k-1 committee members in ± n-1 k-1 ² . Combining the two we have n ± n-1 k-1 ² , possible choices. Part (d): Since all expressions count the same thing they must be equal and we have k ± n k ² = ( n-k + 1) ± n k-1 ² = n ± n-1 k-1 ² . Part (e): We have k ± n k ² = k n ! ( n-k )! k ! = n ! ( n-k )!( k-1)! = n !( n-k + 1) ( n-k + 1)!( k-1)! = ( n-k + 1) ± n k-1 ²...
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