(i) Given: ¹⁵C₈ + ¹⁵C₉ – ¹⁵C₆ – ¹⁵C₇

To prove: ¹⁵C₈ + ¹⁵C₉ – ¹⁵C₆ – ¹⁵C₇=0

We know that:

ⁿC_r = ⁿC_n-r

⇒

⇒¹⁵C₈ + ¹⁵C₉ – ¹⁵C₆ – ¹⁵C₇=¹⁵C₈ + ¹⁵C₉ – ¹⁵C₉ – ¹⁵C₈=0

Hence,proved that ¹⁵C₈ + ¹⁵C₉ – ¹⁵C₆ – ¹⁵C₇=0

(ii) We know that: ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r

Therefore, n = 10 and r = 4

L.H.S = ¹⁰C₄ + ¹⁰C₃ = ¹¹C₄

Hence, proved.