Question

permutations and combinations question
if 6C2r+3 = 6Cr-9 then find 6Cr+1

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

$\displaystyle \sf{ \: }\large{ {}^{6} C_{2r + 3} = \large{ {}^{6} C_{r - 9}} \: }$

TO DETERMINE

$\displaystyle \sf{ \: }\large{ {}^{6} C_{r + 1}}$

FORMULA TO BE IMPLEMENTED

$\displaystyle \sf{ \: }\large{ {}^{n} C_{p} = \large{ {}^{n} C_{q}} \: } \: \: implies \: \: \: p + q = n$

CALCULATION

It is given that

$\displaystyle \sf{ \: }\large{ {}^{6} C_{2r + 3} = \large{ {}^{6} C_{r - 9}} \: }$

$\implies \sf{ 2r + 3 + r - 9 = 6\: }$

$\implies \sf{ 3r - 6 = 6\: }$

$\implies \sf{ 3r = 12\: }$

$\implies \sf{ r = 4\: }$

Hence

$\displaystyle \sf{ \: }\large{ {}^{6} C_{r + 1}}$

$= \displaystyle \sf{ \: }\large{ {}^{6} C_{4 + 1}}$

$= \displaystyle \sf{ \: }\large{ {}^{6} C_{5}}$

$= \sf{ \: 6}$

RESULT

$\boxed{ \: \: \: \displaystyle \sf{ \: }\large{ {}^{6} C_{r + 1}} = 6 \: \: \: }$

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Answer 2

Answer:

(6−r)!

6!

=360

(6−r)r!

6=15

r!=

15

360

=24

r=4