SOLUTION: Please help: (1)Use the binomial theorem to find the 18th term in the binomial expansion of [2m-n(sqrt(2)]^27 (2)Find the 69th number in the 72nd row (n=72) of Pascal’s triangle

Your solution is below.  

Edwin McCravy aka AnlytcPhil

Usually the letter n is used to represent a general number, not a specific
number, but here n is used as a specific number, so I can't speak of the "nth
term", so I'll have to speak of the "pth" term instead to avoid conflict of
letters.

The pth term of , if you start counting with 0 instead of 1, is

, 

where  is the binomial coefficient which is the same as
C(s,p) or sCp or "the number of combinations of s things taken p at a time."

The 18th term is the 17th term if we start counting from 0, so we
substitute  

, , , 

into



 and simplify a little:



Further simplifying the factors separately:
    =  <-- using a TI-84 calculator
    =  = 
    = 
    =  =  =  =  =  = 

Putting them all together:



  <-- answer

------------------------------------

Again we start counting from 0, so the 69th number starting counting with 1,
is the 68th number, when we start counting with 0 instead of 1. 



If you can't use a calculator you can use 

.

  <-- answer

Edwin