Question

Using the concept of Binomial theorem answer the following questions:

a. If the coefficients of x and x² in the expansion of (1+x)^m (1-x)ⁿ  are 3 and -6. Find the value of m and n.

b. Find the total number of terms in (1+x)^m and (1-x)ⁿ.

c. Find the fifth term from the end in the expansion of (1+x)^m and middle term of the expansion (1-x)ⁿ
d. Write the expansion of (1+x)^m.

Answer 1

a. (1 + x)^m (1 − x)ⁿ = (1 + mx + \(\frac{m(m -1)}{2}x^2\) +...) × (1 − nx + \(\frac{n(n -1)}{2}x^2 \) +...)

∴ Coefficient of x = m − n = 3 (given)

∴ Coefficient of x² = \(\frac{n(n-1)}{2} - nm + \frac{m(m - 1)}{2} = -6\) (given)

⇒ n² − n − 2mn + m² − m = −12

⇒ (m − n)² − (m + n) = −12

⇒ (3)² − (m + n) = −12

⇒ m + n = 21

∴ Solving m − n = 3, m + n = 21, we get m = 12, n = 9

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