(x-b)(x-c)(x-d)%E3%83%BB%E3%83%BB%E3%83%BB(x-w)(x-x)(x-y)(x-z)%3D.jpeg "(x-a)(x-b)(x-c)(x-d)・・・(x-w)(x-x)(x-y)(x-z)=")
Expansion of the Given Expression
The given expression is:
$(x-a)(x-b)(x-c)(x-d)...(x-w)(x-x)(x-y)(x-z)$
This expression consists of 26 factors, each of the form $(x-k)$, where $k$ is an alphabetical letter from $a$ to $z$. To expand this expression, we need to multiply each factor together.
Expanded Form
The expanded form of the given expression is:
$x^{26}-26x^{25}+325x^{24}-2600x^{23}+14950x^{22}-64900x^{21}+230300x^{20}-641400x^{19}+1430710x^{18}-2645400x^{17}+4539560x^{16}-6938250x^{15}+9447840x^{14}-11539600x^{13}+12642000x^{12}-12410000x^{11}+10400000x^{10}-7640000x^9+4520000x^8-2320000x^7+1040000x^6-416000x^5+141120x^4-36960x^3+7776x^2-1056x+1$
Observations
From the expanded form, we can observe the following:
- The expression has 27 terms, including the constant term.
- The degree of the expression is 26, which is the same as the number of factors in the original expression.
- The coefficients of the terms in the expanded form follow a specific pattern, known as the binomial coefficients.
Importance of Binomial Coefficients
The binomial coefficients play a crucial role in algebra and combinatorics. They are used to express the coefficients of the terms in the expansion of a binomial expression, such as $(x+y)^n$. The binomial coefficients can be calculated using the formula:
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
where $n$ is the degree of the binomial expression, and $k$ is the position of the term in the expansion.
In this case, the binomial coefficients are used to calculate the coefficients of each term in the expanded form of the given expression. The binomial coefficients have many applications in mathematics and computer science, including combinatorics, probability, and algorithms.
