JEE Main 2020 (Online) 9th January Morning ShiftQuestion 1
The coefficient of $x^4$ in the expansion of $(1+x+x^2)^{10}$ is ________.
Binomial Theorem · Multinomial Expansion
Answer$615$
Solution
Concept. Choose how many factors give $x^2$ and how many give $x$; the rest give $1$. Formulas used. (1) Number of ways to choose $a$, $b$, $c$ factors out of $10$: $\dfrac{10!}{a!\,b!\,c!}$Step 1 - Condition. If $a$ factors give $x^2$ and $b$ give $x$, then $2a+b=4$: $(a,b)=(0,4),(1,2),(2,0)$. Step 2 - Each case. By (1): $$(0,4):\ {}^{10}C_4=210,\qquad(1,2):\ \frac{10!}{1!\,2!\,7!}=360,\qquad(2,0):\ {}^{10}C_2=45.$$ Step 3 - Add. $210+360+45=615$. Answer: 615