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Greatest Binomial Coefficients

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Question

Simplify : $^{34} C_{5} + 4 \sum r = 0^{38 - r} C_{4}$

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Solution

$^{34} C_{5} + 4 \sum r = 0$ $^{38 - r} C_{4}$
$=^{34} C_{5} +^{38} C_{4} +^{37} C_{4} +^{36} C_{4} +^{35} C_{4} +^{34} C_{4}$
$= (^{34} C_{5} +^{34} C_{4}) +^{35} C_{4} +^{36} C_{4} +^{37} C_{4} +^{38} C_{4}$
$=^{35} C_{5} +^{35} C_{4} +^{36} C_{4} +^{37} C_{4} +^{38} C_{4}$ (Using $^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$ )
$=^{36} C_{5} +^{36} C_{4} +^{37} C_{4} +^{38} C_{4}$
$=^{37} C_{5} +^{37} C_{4} +^{38} C_{4}$
$=^{38} C_{5} +^{38} C_{4}$
$=^{39} C_{5}$

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