Binomial Expansion and Pascal Triangle

1. Write down first 14 rows of the Pascal Triangle
2. Prove that ⁿC_r = ⁿC_n-r
3. Calculate
   • ⁵C₃+⁵C₄
   • ⁸C₂+2⁸C₃+⁸C₄
   • ⁹C₄+3⁹C₅+3⁹C₆+⁹C₇
   • ⁷C₂+4⁷C₃+6⁷C₄+4⁷C₅+⁷C₆
4. By comparing your results in above questions with some other binomial coefficients try to find the following sums
   • ⁿC_r-1+ⁿC_r
   • ⁿC_r-2+2ⁿC_r-1+ⁿC_r
   • ⁿC_r-3+3ⁿC_r-2+3ⁿC_r-1+ⁿC_r

   Prove the above results

5. There is a relationship between (k+1) successive coefficients in the n^th row of the Pascal Triangle with a coefficient in the (n+k)^th row. Find this relationship.
6. Show that your formula (the relationship that you found above) works for the sum _r=0∑^r=8(⁸C_r)(²⁷C_8-r)
7. By comparing the coefficients of (1+x)ⁿ(1+x)^k and (1+x)^n-k prove your formula in Q5.