#Polynomial expansion

In the expansion ( a + b + c + d ) 500 (a+b+c+d)^{500} the coefficients of a 200 b 50 c 170 d 80 a^{200}b^{50}c^{170}d^{80} and a 198 b 52 c 171 d 79 a^{198}b^{52}c^{171}d^{79} are respectively X X and Y Y . If X Y = 1 0 n \frac{X}{Y}=10^n enter n |n|


The answer is 0.8464.

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1 solution

Rafsan Rcc
May 5, 2021

The given expression is ( a + b + c + d ) 500 (a+b+c+d)^{500} . Here we take each term of the polynomial and multiply it by all the ones of the other one, and so on. Now we want to find the coefficient of one of the terms. To do this we notice the exponents of each of the variables. Like ,for a 200 b 50 c 170 d 80 a^{200}b^{50}c^{170}d^{80} we need to take a 200 times, b 50 times , c 170 times and d 80 times. In the expansion we have 500 of each of them. Now we need to consider the number of possible permutations and that number will be the required coefficient. Now we can arrange 200 a’s, 50 b’s, 170 c’s and 80 d’s in 500 ! 200 ! × 50 ! × 170 ! × 80 ! \frac{500!}{200!×50!×170!×80!} ways and so X = 500 ! 200 ! × 50 ! × 170 ! × 80 ! X=\frac{500!}{200!×50!×170!×80!}

Similarly Y = 500 ! 198 ! × 52 ! × 171 ! × 79 ! Y=\frac{500!}{198!×52!×171!×79!} X Y = 198 ! × 52 ! × 171 ! × 79 ! 200 ! × 50 ! × 170 ! × 80 ! = 51 × 52 × 171 199 × 200 × 80 0.1424 = 1 0 n ∴\frac{X}{Y}=\frac{198!×52!×171!×79!}{200!×50!×170!×80!}=\frac{51×52×171}{199×200×80}≈0.1424=10^n n = l o g 0.1424 0.8464 ∴|n|=|log⁡0.1424 |≈\boxed{0.8464}

Great explanation. Just for reference, this is called the multinomial theorem .

Chris Lewis - 1 month, 1 week ago

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