How to Simplify (11-1)^9 Using Binomial Expansion?

[mr_coffee]

Hello everyone.

I'm studying for my exam and I'm reveiwing some problems but this one isn't making sense to me:

11^9(9 choose 0) - 11^8(9 choose 1) + 11^7(9 choose 2) - ... - 11^2(9 choose 7) + 11^1 (9 choose 8) - 11^0 (9 choose 9)


answer:
(11-1)^9 = 10^9 = 1,000,000,000.

work looked like:
(9 choose 0) 11^9 (-1)^0 + (9 choose 1)11^8(-1)^1


Can someone explain to me how they did this?
I see the 11 is decreasing, and the signs are alternating, it looks like a binomial expansion but I'm not seeing how they simplified that.

Thanks

 

[Office_Shredder]

It IS a binomial expansion. It's 11^k*(-1)^9-k*₉choose_k with k going from 0 to 9

Try expanding (11-1)⁹ and compare them

 

[mr_coffee]

So would
a = 11
and
b = -1
and
n = 9

(11-1)^9 = (9 choose 0)11^9 + (9 choose 1)11^8*(-1) +(9 choose 2)11^7*(-1)^2...

I see this is going to work, from the written out expansion, i can see n = 9, b = -1, a = 11,


ahh i got it@!

thank u!