Here we can find the coefficient of x4 in the given expression. We know that Binomial theorem for Positive Integral index If a, b, are any two real numbers and n any natural number then (a+b)n =
n
‌
C0anb° +
n
‌
C1an−1b +
n
‌
C2an−2b2 + ... +
n
‌
Cran−rbr + ... +
n
‌
Cnx0bn , where
n
‌
Cr =
n!
(n−r)!r!
Here
n
‌
C0,
n
‌
C1,
n
‌
C2 ...
n
‌
Cn are called binomial coefficients i.e., (a+b)n =
n
Σ
r=0
n
‌
Cran−rbr Given (1+x+x2)3 Considering the expansion of (1+x+x2)3 in the form ((1+x)+x2)3 We have (1+x)3+3(1+x)2x2+3(1+x)x4+x6. (Since, by definition of binomial theorem) Here x4 comes in the second and third term only. Adding its coefficients, 3 + 3 = 6. Therefore, The coefficient of x4 in the expansion of (1+x+x2)3 is 6.