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Combination

Simplify 34...

Question

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

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Solution

$^{34} C_{5} + \sum_{r = 0}^{4}^{38 - r} C_{4}$
$=^{34} C_{5} +^{38} C_{4} +^{37} C_{4} +^{36} C_{4} +^{35} C_{4} +^{34} C_{4}$
$= (^{34} C_{5} +^{34} C_{4}) +^{35} C_{4} +^{36} C_{4} +^{37} C_{4} +^{38} C_{4}$
$=^{35} C_{5} +^{35} C_{4} +^{36} C_{4} +^{37} C_{4} +^{38} C_{4} (∵^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r})$
$=^{36} C_{5} +^{36} C_{4} +^{37} C_{4} +^{38} C_{4}$
$=^{37} C_{5} +^{37} C_{4} +^{38} C_{4}$
$=^{38} C_{5} +^{38} C_{4}$
$=^{39} C_{5}$
$= \frac{39!}{34! \times 5!}$
$= \frac{39 \times 38 \times 37 \times 36 \times 35}{5 \times 4 \times 3 \times 2 \times 1} (∵^{n} C_{r} = \frac{n!}{r! (n - r)!})$
$= 575757$
Hence the value of $^{34} C_{5} + \sum_{r = 0}^{4}^{38 - r} C_{4}$ is $^{39} C_{5}$ or $575757$ .

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