NCr

Definition

The term $nCr$ represents the number of ways to choose $r$ items from a set of $n$ items, without regard to order. It is a fundamental concept in the Binomial Theorem, which describes the expansion of binomial expressions.

5 Must Know Facts For Your Next Test

1. $nCr$ is calculated using the formula: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$, where $n!$ represents the factorial of $n$.
2. The value of $nCr$ represents the number of unique subsets of size $r$ that can be formed from a set of $n$ items.
3. The Binomial Theorem states that $(a + b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r$, where $nCr$ appears as the coefficient of each term.
4. The values of $nCr$ can be organized in a triangular array known as Pascal's Triangle, where each number is the sum of the two numbers above it.
5. The concept of $nCr$ is widely used in various areas of mathematics, including probability, combinatorics, and discrete mathematics.

Review Questions

• Explain the relationship between the Binomial Theorem and the term $nCr$.
  • The term $nCr$ is a fundamental component of the Binomial Theorem, which describes the expansion of binomial expressions of the form $(a + b)^n$. The Binomial Theorem states that the coefficient of the term $a^{n-r}b^r$ in the expansion is given by $nCr$, which represents the number of ways to choose $r$ items from a set of $n$ items. This connection between the Binomial Theorem and the combinatorial concept of $nCr$ is crucial for understanding the structure and properties of binomial expansions.
• Describe how the formula for calculating $nCr$ is derived and explain its significance.
  • The formula for calculating $nCr$ is $\binom{n}{r} = \frac{n!}{r!(n-r)!}$, where $n!$ represents the factorial of $n$. This formula is derived by considering the number of ways to arrange the $r$ selected items (given by $r!$) and the number of ways to arrange the remaining $n-r$ items (given by $(n-r)!$). Dividing the total number of permutations ($n!$) by the number of permutations that represent the same combination gives the final formula for $nCr$. This formula is significant because it provides a concise and efficient way to calculate the number of combinations, which is a crucial concept in various areas of mathematics, including probability, combinatorics, and discrete mathematics.
• Analyze the properties of $nCr$ and explain how they contribute to the structure and applications of the Binomial Theorem.
  • The properties of $nCr$ play a crucial role in the structure and applications of the Binomial Theorem. For instance, the symmetry property of $nCr$, where $nCr = nC(n-r)$, reflects the symmetry observed in the expansion of $(a + b)^n$. Additionally, the recursive nature of $nCr$, where $nCr = (n-1)C(r-1) + (n-1)Cr$, allows for the efficient computation of binomial coefficients and the construction of Pascal's Triangle. These properties, combined with the formula for $nCr$, enable the Binomial Theorem to be a powerful tool in various mathematical disciplines, such as probability, combinatorics, and discrete mathematics, where the enumeration of combinations is essential.

Related terms

Combination: A combination is a way of selecting items from a set, where the order of selection does not matter.

Permutation: A permutation is a way of arranging items in a specific order, where the order of selection does matter.

Binomial Coefficient: The binomial coefficient $\binom{n}{r}$ is another way to denote $nCr$, the number of combinations of $r$ items from a set of $n$ items.