Question

Find the eighth term in (a+b) ^14

Answer 1

`(a+b)^n\\\displaystyle T_r=\binom{n}{r-1}a^(n-r+1)b^(r-1)\\\\T_8=\binom{14}{8-1}a^(14-8+1)\cdot b^(8-1)\\T_8=\binom{14}{7}a^7b^7\\T_8=(14!)/(7!7!)\cdota^7b^7\\T_8=(8\cdot9\cdot10\cdot11\cdot12\cdot13\cdot14)/(2\cdot3\cdot4\cdot5\cdot6\cdot7)\cdot a^7b^7\\T_8=3432a^7b^7`

Answer 2

the nth term in the expansion (a+b)^m=
(
`((m)!)/((n-1)!(m-(n-1))!)`)a^(m-(n-1))b^(n-1)

so

8th term
(
`((14)!)/((8-1)!(14-(8-1))!)`)a^(14-(8-1))b^(8-1)=
(
`((14)!)/(7!(7)!)`)a^(7)b^(7)=
(
`(14*13*12*11*10*9*8)/(7!)`)a^(7)b^(7)=
3432a⁷b⁷ is answer