Binomial expansion

Probability Level 1

(1) \quad What is the number of the terms in the expansion of ( 1 + x ) 5 (1+x)^5 ?

(2) \quad What is the value of the expansion in the question above if we substitute x = 2 x=2 ?

6, 244 7, 243 6, 243 5, 244

Tahmid Chowdhury by Tahmid Chowdhury , 23, Bangladesh

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2 solutions

Sabhrant Sachan
May 1, 2016

Relevant wiki: Binomial Theorem

The Number of Terms in ( 1 + x ) n is n + 1 , For Example : 1. ( 1 + x ) 1 = 1 + x ⟹ 2 terms 2. ( 1 + x ) 2 = 1 + 2 x + x 2 ⟹ 3 terms 3. ( 1 + x ) 3 = 1 + 3 x + 3 x 2 + x 3 ⟹ 4 terms ... and So on If we substitute x=2 , we get ( 1 + 2 ) 5 = 3 5 = 243 \text {The Number of Terms in } (1+x)^n \text { is } n+1 , \text { For Example : }\\ \text{1. }(1+x)^1=1+x \implies \text { 2 terms} \\ \text{2. }(1+x)^2=1+2x+x^2 \implies \text { 3 terms}\\ \\ \text{3. }(1+x)^3=1+3x+3x^2+x^3 \implies \text { 4 terms ... and So on}\\ \text{If we substitute x=2 , we get } (1+2)^5 = 3^5 = \boxed{243}

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Tahmid Chowdhury
Apr 29, 2016

(1+x)^5=1+5x+10x^2+10x^3+5x^4+x^5 Putting value of x=2 we get from the expression= 1+5(2)+10(2)^2+10(2)^3+5(2)^4+(2)^5=243 Total number of terms in the expansion (n+1)=(5+1)=6

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