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nCr Definitions and Properties

If n C 4= n C...

Question

If $^{n} C_{4} =^{n} C_{5}$ , find $^{n} C_{3}$

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Solution

$^{n} C_{x} =^{n} C_{y} \Rightarrow x + y = n o r x = y$
Here, 4 is not equal to 4. Another possibility is n=4+5=9
Therefore $^{n} C_{3} =^{9} C_{3}$
= $\frac{9!}{3! \times 6!}$

= 84

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Similar questions

a =^{n} C_{0} +^{n} C_{3} +^{n} C_{6} + . . .

b =^{n} C_{1} +^{n} C_{2} +^{n} C_{4} +^{n} C_{5} +^{n} C_{7} +^{n} C_{8} + . . .

c =^{n} C_{1} -^{n} C_{2} +^{n} C_{4} -^{n} C_{5} +^{n} C_{7} -^{n} C_{8} + . . .

,
then the value of

{(a - \frac{b}{2})}^{2} + \frac{3 c^{2}}{4}

a =^{n} C_{0} +^{n} C_{3} +^{n} C_{6} + . . .

b =^{n} C_{1} +^{n} C_{2} +^{n} C_{4} +^{n} C_{5} +^{n} C_{7} +^{n} C_{8} + . . .

c =^{n} C_{1} -^{n} C_{2} +^{n} C_{4} -^{n} C_{5} +^{n} C_{7} -^{n} C_{8} + . . .

,
then the value of

{(a - \frac{b}{2})}^{2} + \frac{3 c^{2}}{4}

^{n} C_{4}

^{n} C_{5}

^{n} C_{6}

(1 + x)^{n}

are in A.P. then

n =

^{n} C_{3} =^{n} C_{5^{'}}

then find the value of n:

^{n} C_{3} +^{n} C_{4} >^{n + 1} C_{3}

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nCr Definitions and Properties

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