Simplify: \(^{n}C_{r} ÷ ^{n}C_{r-1}\)
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Correct Answer: Option D
Explanation:
\(^{n}C_{r} = \frac{n!}{(n-r)! r!}\)
\(^{n}C_{r - 1} = \frac{n!}{(n - (r - 1))! (r - 1)!}\)
\(^{n}C_{r} ÷ ^{n}C_{r - 1} = \frac{n!}{(n - r)! r!} ÷ \frac{n!}{(n-(r-1))!(r-1)!}\)
= \(\frac{n!}{(n-r)! r!} \times \frac{(n-(r-1)! (r-1)!}{n!}\)
= \(\frac{(n + 1 - r)! (r - 1)!}{(n - r)! r!}\)
= \(\frac{(n+1-r)(n-r)! (r-1)!}{(n-r)! r (r - 1)!}\)
= \(\frac{n + 1 - r}{r}\)
\(^{n}C_{r} = \frac{n!}{(n-r)! r!}\)
\(^{n}C_{r - 1} = \frac{n!}{(n - (r - 1))! (r - 1)!}\)
\(^{n}C_{r} ÷ ^{n}C_{r - 1} = \frac{n!}{(n - r)! r!} ÷ \frac{n!}{(n-(r-1))!(r-1)!}\)
= \(\frac{n!}{(n-r)! r!} \times \frac{(n-(r-1)! (r-1)!}{n!}\)
= \(\frac{(n + 1 - r)! (r - 1)!}{(n - r)! r!}\)
= \(\frac{(n+1-r)(n-r)! (r-1)!}{(n-r)! r (r - 1)!}\)
= \(\frac{n + 1 - r}{r}\)