What is the value of adding the combinations of 11 choose 1, 11 choose 3, and so on up to 11 choose 11?

To select an odd number of elements from a group of eleven, you always leave an even number of elements behind. This means that the sum of combinations of 11 elements taken at odd positions is equal to the sum of combinations taken at even positions. Therefore, the number of possible ways to choose a subset of eleven elements is twice the number of ways to select an odd number of elements. Hence, there are 1024 ways to choose an odd number of elements, which is half of the total number of ways (2^11).


Solution 1:

There may be quicker ways, but one method is to apply the binomial theorem twice. First, we have $(1+1)^{11}=sumlimits_{i=0}^{11}binom{11}{i}$, and then $(1-1)^{11}=sumlimits_{i=0}^{11}binom{11}{i}(-1)^i$ which we can subtract from the first result. Dividing the difference by two gives us the answer.


Solution 2:


The sum of the binomial coefficients for choosing an odd number of elements out of a set of eleven is equal to the sum of the binomial coefficients for choosing an even number of elements. This is because selecting an odd number of elements always leaves behind an even number of elements in the set of eleven.

The number of ways to select an odd number of elements from a set of eleven is half the number of ways to select any subset of eleven elements.

To select a subset, we must either include or exclude each element, resulting in $2^{11}$ possibilities.

The total count of selecting an odd number of elements is equivalent to choosing half of the total number of elements which is equal to $frac{1}{2} cdot 2^{11}$. This simplifies to $2^{10}$, resulting in a total count of 1024.

The aforementioned logic is invalid for cases where the main set contains an even number of elements. However, the principle that the number of ways to select an odd number of elements is equal to half the total number of subsets remains valid. Are you able to comprehend why this is so?


Solution 3:


The identities $binom{11}{1}=binom{11}{10}$ and $binom{11}{3}=binom{11}{8}$, among others, allow us to express the sum $binom{11}{1}+binom{11}{3}+ldots+binom{11}{11}$ as $binom{11}{10}+binom{11}{8}+ldots+binom{11}{0}$. Both of these expressions are equal to half the sum $binom{11}{0}+binom{11}{1}+binom{11}{2}+ldots+binom{11}{11}$, which equals $2^{11}$. Therefore, $2^{10}=color{red}{1024}$.

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