Simplify: {{}ⁿC_{r}}{{}ⁿC_{r-1}}

Correct Answer is : n+1−rrn+1-r/rrn+1−r

Correct Answer is : n+1−rrn+1-r/rrn+1−r

Test Index

WAEC Further Mathematics 2015 Paper

Question : 34

Total: 40

Simplify:

\frac{{}^{n}C_{r}}{{}^{n}C_{r-1}}

Solution:

Concept:

Use the standard combination formula

^{n}C_{k} = \frac{n!}{(n-k)!\,k!}

and simplify the ratio by cancelling factorial terms.

Explanation:

Write the numerator and denominator using the formula.

^{n}C_{r} = \frac{n!}{(n-r)!\,r!}

^{n}C_{r-1} = \frac{n!}{(n-(r-1))!\,(r-1)!} = \frac{n!}{(n+1-r)!\,(r-1)!}

Now take the ratio.

\frac{^{n}C_{r}}{^{n}C_{r-1}} = \frac{n!}{(n-r)!\,r!} \div \frac{n!}{(n+1-r)!\,(r-1)!}

Change division to multiplication by the reciprocal.

= \frac{n!}{(n-r)!\,r!} \times \frac{(n+1-r)!\,(r-1)!}{n!}

Cancel

n!

and expand

(n+1-r)!

(n+1-r)(n-r)!

= \frac{(n+1-r)(n-r)!\,(r-1)!}{(n-r)!\,r\,(r-1)!}

Cancel the common factors

(n-r)!

and

(r-1)!

= \frac{n+1-r}{r}

Answer:

\frac{n+1-r}{r}

, which is Option D.

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