Evaluate: $^{15} C_{8} +^{15} C_{9} -^{15} C_{6} -^{15} C_{7}$

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Solution

Given: $^{15} C_{8} +^{15} C_{9} -^{15} C_{6} -^{15} C_{7}$

Using $^{n} C_{r} =^{n} C_{n - r}$ ,
$^{15} C_{8} +^{15} C_{9} =^{15} C_{7} +^{15} C_{6}$ ----- (1)
Now from question,
$^{15} C_{8} +^{15} C_{9} -^{15} C_{6} -^{15} C_{7}$
$=^{15} C_{7} +^{15} C_{6} -^{15} C_{6} -^{15} C_{7})$
$= 0$
Hence, $(^{15} C_{8} +^{15} C_{9}) - (^{15} C_{6} +^{15} C_{7}) = 0$

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Evaluate: 15C8+15C9−15C6−15C7

Given: 15C8+15C9−15C6−15C7Using nCr=nCn−r,15C8+15C9=15C7+15C6 ----- (1)Now from question,15C8+15C9−15C6−15C7=15C7+15C6−15C6−15C7)=0Hence, (15C8+15C9)−(15C6+15C7)=0

Evaluate: $^{15} C_{8} +^{15} C_{9} -^{15} C_{6} -^{15} C_{7}$