We know, n C r + n C r-1 = n+1 C r Now, we have, 56-r C 3 + 50 C 4 = 55 C 3 + 54 C 3 + 53 C 3 + 52 C 3 + 51 C 3 + 50 C 3 + 50 C 4 = 55 C 3 + 54 C 3 + 53 C 3 + 52 C 3 + 51 C 3 + 51 C 4 = 55 C 3 + 54 C 3 + 53 C 3 + 52 C 3 + 52 C 4 = 55 C 3 + 54 C 3 + 53 C 3 + 53 C 4 = 55 C 3 + 54 C 3 + 54 C 4 = 55 C 3 + 55 C 4 = 56 C 4

Write the value of .

We know,

ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r

Now, we have,

^56-rC₃ + ⁵⁰C₄

= ⁵⁵C₃ + ⁵⁴C₃ + ⁵³C₃ + ⁵²C₃ + ⁵¹C₃ + ⁵⁰C₃ + ⁵⁰C₄

= ⁵⁵C₃ + ⁵⁴C₃ + ⁵³C₃ + ⁵²C₃ + ⁵¹C₃ + ⁵¹C₄

= ⁵⁵C₃ + ⁵⁴C₃ + ⁵³C₃ + ⁵²C₃ + ⁵²C₄

= ⁵⁵C₃ + ⁵⁴C₃ + ⁵³C₃ + ⁵³C₄

= ⁵⁵C₃ + ⁵⁴C₃ + ⁵⁴C₄

= ⁵⁵C₃ + ⁵⁵C₄

= ⁵⁶C₄