Write the value of

We know:

nCr-">−-1 + nCr = n+1Cr

Now, we have:

$\sum_{r=1}^{6}{ }^{56-r} C_{3}+{ }^{50} C_{4}$

$={ }^{55} C_{3}+{ }^{54} C_{3}+{ }^{53} C_{3}+{ }^{52} C_{3}+{ }^{51} C_{3}+{ }^{50} C_{3}+{ }^{50} C_{4}$

$={ }^{55} C_{3}+{ }^{54} C_{3}+{ }^{53} C_{3}+{ }^{52} C_{3}+{ }^{51} C_{3}+{ }^{51} C_{4}$

$={ }^{55} C_{3}+{ }^{54} C_{3}+{ }^{53} C_{3}+{ }^{52} C_{3}+{ }^{52} C_{4}$

$={ }^{55} C_{3}+{ }^{54} C_{3}+{ }^{53} C_{3}+{ }^{53} C_{4}$

$={ }^{55} C_{3}+{ }^{54} C_{3}+{ }^{54} C_{4}$

$={ }^{55} C_{3}+{ }^{55} C_{4}$

$={ }^{56} C_{4}$