Question

Find the terms. Leave coefficients as factorials


term with t^5 in (v-t)^17

term with x^10 in (x+y)^19

Answer 1

for (a+b)^n
the general term t_r+1 = C(n,r) a^(n-r) b^r

so term t_r+1 = C(17,r) v^(n-r) t)^r

so (-t)^r ----> t^5
r = 5


so it must be term(6) and it is C(17,5) v^5 (-t)^12
= - 6188 v^12 t^5

check:
(v-t)^17
=C(17,0) v^17 + C(17,1) v^16 (-t) + C(17,2) v^15 (-t)^2 + ... + C(17,5) v^12 (-t)^5 + ...
= v^17 - 17v^16 t + 136 v^15 t^2 + ... - 6188 v^12 t^5 + ...

do the 2nd the same way