Question

what is the value of 6C0+6C1+6C2+6C3+ 6C4+6C5+6C6?​

Answer 1

C0.12C6−6C1.11C6+6C2.10C6+6C2.10C6−6C3.9C6+6C4.8C6−6C5.7C6+6C6.6C6

1×924−6×462+15×210+15×210−20×84+15×28−6×7+1×1

924−2772+3150+3150−1680−420−41+1

2311

Crystii

Thanks for good question

Answer 2

The value of the given sum series of combinations with n=6 is 64.

Combinations ( $C_{r} ^{n}$ ) :

• $C_{r} ^{n}$ denotes combinations of selections
• It shows the number of ways we can select "r" items from a set of total "n" items
• The formula for finding it is :

                   $C_{r} ^{n} = (n!) / (n-r)! * (r!)$

                   where $n! = n * (n-1) * (n-2) * .... 2*1$

The formula for summation of $C_{r} ^{n}$ is as follows :

        $C_{n} ^{n} + C_{n-1} ^{n} + C_{n-2} ^{n} + .... + C_{2} ^{n} + C_{1} ^{n} + C_{0} ^{n} = 2^{n}$

        ∑$C_{r} ^{n} (r = (0, n) )= 2^{n}$

The given series of finding the sum of combinations resembles the above generic sum series of combinations. For the given question, n is 6 and the combination is summed for values of r = 0, 1, 2, 3, 4 ,5 and 6.

Here, we are required to find the sum of the series :

                      $= C_{0} ^{6} + C_{1} ^{6} + C_{2} ^{6} + C_{3} ^{6} + C_{4} ^{6} + C_{5} ^{6} + C_{6} ^{6}$

                      $= C_{r} ^{n} (r = (0, 6) )$

                      $= 2^{6}$. ( using the summation formula defined above )

                      $= 64$

Thus, the sum of the given combination sum series is 64.

To learn more about Combination, visit

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