The value of ∑r=06(6Cr ⋅ 6C6−r) is equal to

∑R=06Cr 6.C6−r 6 =C0 6.C6 6+C1 6.C5 6+.......+C6 6.C0 6 Now, (1+x)6(1+x)6 =(C0 6+C1 6x+C2 6x2+....+C6 6x6) (C0 6+C1 6x+C2 6x2+....+C6 6x6) Comparing coefficient of both sides C0 6.C6 6+C1 6+C5 6+.......+C6 6.C0 6=C6 12 =924

The value of ∑r=06(6Cr ⋅ 6C6−r) is equal to

∑R=06Cr 6.C6−r 6 =C0 6.C6 6+C1 6.C5 6+.......+C6 6.C0 6 Now, (1+x)6(1+x)6 =(C0 6+C1 6x+C2 6x2+....+C6 6x6) (C0 6+C1 6x+C2 6x2+....+C6 6x6) Comparing coefficient of both sides C0 6.C6 6+C1 6+C5 6+.......+C6 6.C0 6=C6 12 =924

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The value of $\sum_{r = 0}^{6} (^{6} C_{r} \cdot^{6} C_{6 - r})$ is equal to

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1024

1124

924

1324

answer is D.

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Detailed Solution

$\sum_{R = 0}^{6} {}^{6}C_{r} . {}^{6}C_{6 - r}$ $= {}^{6}C_{0} . {}^{6}C_{6} + {}^{6}C_{1} . {}^{6}C_{5} + ....... + {}^{6}C_{6} . {}^{6}C_{0}$

Now,

${(1 + x)}^{6} {(1 + x)}^{6} = ({}^{6}C_{0} + {}^{6}C_{1} x + {}^{6}C_{2} x^{2} + .... + {}^{6}C_{6} x^{6}) ({}^{6}C_{0} + {}^{6}C_{1} x + {}^{6}C_{2} x^{2} + .... + {}^{6}C_{6} x^{6})$
Comparing coefficient of both sides
${}^{6}C_{0} . {}^{6}C_{6} + {}^{6}C_{1} + {}^{6}C_{5} + ....... + {}^{6}C_{6} . {}^{6}C_{0} = {}^{12}C_{6} = 924$