Answer 1

We know:
ⁿC_r  \[-\]₁ + ⁿC_r = ⁿ⁺¹C_r

\[\text{Now, we have}: \]
\[ \sum^6_{r = 1} {}^{56 - r} C_3 + {}^{50} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{53} C_3 +^{52} C_3 +^{51} C_3 +^{50} C_3 +^{50} C_4\]

\[=^{55} C_3 +^{54} C_3 +^{53} C_3 +^{52} C_3 +^{51} C_3 +^{51} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{53} C_3 +^{52} C_3 +^{52} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{53} C_3 +^{53} C_4 \]
\[ =^{55} C_3 +^{54} C_3 +^{54} C_4 \]
\[ =^{55} C_3 +^{55} C_4 \]
\[ =^{56} C_4 \]