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  3. 조합론 Combinatorics

Binomial theorem, the exponent value of the term with the largest coefficient when developing (aX+bY)^n

작성자 Uploader : anyway 작성일 Upload Date: 2019-08-19변경일 Update Date: 2019-10-21조회수 View : 349

When expanding (aX+bY)^n, find the k value with which the coefficient of X^(k)*Y^(n-k) is the largest.

If a ≥ 1, b ≥ 1 and n ≥ 1 (natural number),

The coefficient of X^(k)*Y^(n-k) is :

nCk*a^(k)*b^(n-k)

This coefficient increases as k increases, then reaches the maximum and decreases.

Let r = k+1, then the coefficient of the next term is :

nCr*a^(r)*b^(n-r)

Among two terms, the former term is larger than or equal to the latter. Thus,  

nCk*a^k*b^(n-k) ≥ nCr+1*a^(r)*b*(n-r)

Dividing by a^k*b^(n-k) (> 0),

nCk ≥ nCr*a/b

nCk/nCr ≥ a/b

(n!/((n-k)!*k!))/(n!/((n-r)!*r!)) ≥ a/b

((n-r)!*r!)/((n-k)!*k!)) ≥ a/b

Since r = k+1,

((n-k-1)!*(k+1)!)/((n-k)!*k!)) ≥ a/b

(k+1)/(n-k) ≥ a/b

k+1 ≥ (a/b)*n - (a/b)*K

(1+a/b)*k ≥ (a/b)*n -1

k ≥ ((a/b)*n-1)/(1+a/b)

Res = ((a/b)*n-1)/(1+a/b)

Then, k ≥ Res (natural number)


*** 참고문헌[References] ***
Res = ((a/b)*n-1)/(1+a/b)
변수명 Variable 변수값 Value 변 수 설 명 Description of the variable


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