The coefficient of $x^{4}$ in the expansion of the ${(1 + x + x^{2} + x^{3})}^{10}$

720

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710

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705

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700

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Solution

The correct option is C $705$
${(1 + x + x^{2} + x^{3})}^{10} = {[(1 + x) + x^{2} (1 + x)]}^{10} = {(1 + x)}^{10} {(1 + x^{2})}^{10} = \sum^{10} C_{r_{1}} x^{r_{1}}^{10} C_{r_{2}} x^{2 r_{2}} = \sum^{10} C_{r_{1}}^{10} C_{r_{2}} x^{r_{1} + 2 r_{2}}$

For the coefficient of $x^{4}$ , $r_{1} + 2 r_{2} = 4$
There will be three cases:
$(i) r_{1} = 4, r_{2} = 0, (ii) r_{1} = 2, r_{2} = 1, (iii) r_{1} = 0, r_{2} = 2,$

The coefficient of $x^{4}$ in the expansion is $^{10} C_{4} \times^{10} C_{0} +^{10} C_{2} \times^{10} C_{1} +^{10} C_{0} \times^{10} C_{2} = 705$

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