The value of \(\left({ }^{7} \mathrm{C}_{0}+{ }^{7} \mathrm{C}_{1}\right)+\left({ }^{7} \mathrm{C}_{1}+{ }^{7} \mathrm{C}_{2}\right)+\ldots \ldots+\) \(\left({ }^{7} \mathrm{C}_{6}+{ }^{7} \mathrm{C}_{7}\right)\) is (1) \(2^{8}-2\) (2) \(2^{8}-1\) (3) \(2^{8}+1\) (4) \(2^{8}\) (5) \(2^{7}-2\)

Binomial coefficients are an essential part of combinatorics and are represented as \(\binom{n}{k}\). This notation expresses the number of ways to choose \('k'\) elements from a set of \('n'\) elements without regard to the order of selection.
Binomial coefficients can be calculated using the formula \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
where \( n! \) (n factorial) represents the product of all positive integers up to \( n \). Here are some key points about binomial coefficients:
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• They symmetric, meaning \( \binom{n}{k} = \binom{n}{n-k} \).
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• They are used in the expansion of binomial expressions.
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• They appear in Pascal's triangle where each entry is the sum of the two numbers directly above it.
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Understanding binomial coefficients helps in solving problems related to combinations and binomial expansions.

The binomial expansion involves expanding expressions of the form \((a + b)^n\).
The Binomial Theorem provides a formula for this expansion:
\[ (a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1} b^1 + \binom{n}{2}a^{n-2} b^2 + \text{...} + \binom{n}{n}a^0 b^n \]
This theorem shows how a binomial expression can be expanded into a sum involving terms of the form \( a^k b^{n-k} \). Each term in the expansion is multiplied by a binomial coefficient, \( \binom{n}{k} \). Here's why this is useful:
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• It simplifies the process of expanding polynomials.
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• It helps in calculating probabilities, especially in binomial distributions.
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• It is foundational in algebra and calculus.
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For example, consider \( (1 + x)^7 = \binom{7}{0}1^7x^0 + \binom{7}{1}1^6x^1 + \text{...} + \binom{7}{7}1^0x^7 \). The coefficients here are the binomial coefficients, summing to create the expanded form.

Combinatorics is the branch of mathematics dealing with combinations, permutations, and the counting of objects.
It provides tools for solving problems that involve discrete structures and is vital in fields such as computer science, cryptography, and statistics.
Here are some foundational concepts in combinatorics:
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• Permutations: Arrangements of objects in a specific order. Calculated using \( n! \) for \( n \) distinct objects.
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• Combinations: Selection of objects without regard to order. Involves binomial coefficients, as seen in the formula \( \binom{n}{k} \).
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• Principle of Inclusion-Exclusion: A method used to count the number of elements in overlapping sets.
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• Graph Theory: Studies graphs and networks to solve connectivity problems and optimize paths.
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Understanding combinatorics allows you to tackle complex problems involving counting and arrangements. In the exercise, you used combinatorial principles to simplify the sum of binomial coefficients and arrive at the final solution.