The Binomial Expansion Question?

You have only given part of the question, you must require a particular term.

The term involving x^ry^(26-r) is 26Crx^ry^(26-r) where 26Cr=26!/(r!(26-r)!).

These coefficients can be obtained on a calculator from the nCr button.

Yes, remember when you grew beyond school-boy maths to college level maths you were asked to multiply out (a+b)^4 or something like that so that after working really hard at getting your answer and simplifying it you get taught Pascal's triangle, then you feel really cleaver 'cause you can multiply out (a+b)^n a lot quicker.

Well now you're being introduced to the same thing, you're expected to write out Pascal's triangle upto (x+y)^26 and take ages and lots and lots of paper (or blackboard/whiteboard space) so that you can be told an easier way of doing it.

The easier way to do it is called Binomial theorem, it was proposed by Euclid but took until Blaise Pascal to be actually proved. Hopefully you know enough probability and statistics to know how to do n choose k.

(x+y)^n =

∞

∑ (n choose k) x^k y^(n-k)

k=0

The answer by the way is:

x^26+26 x^25 y+325 x^24 y^2+2600 x^23 y^3+14950 x^22 y^4+65780 x^21 y^5+230230 x^20 y^6+657800 x^19 y^7+1562275 x^18 y^8+3124550 x^17 y^9+5311735 x^16 y^10+7726160 x^15 y^11+9657700 x^14 y^12+10400600 x^13 y^13+9657700 x^12 y^14+7726160 x^11 y^15+5311735 x^10 y^16+3124550 x^9 y^17+1562275 x^8 y^18+657800 x^7 y^19+230230 x^6 y^20+65780 x^5 y^21+14950 x^4 y^22+2600 x^3 y^23+325 x^2 y^24+26 x y^25+y^26

Just remember this whole ridiculous exercise is to get you to waste loads of time so that you appreciate why you need to know binomial theorem so even though you have the answer, I strongly recommend you take a couple of big blackboards or whiteboards and write out Pascal's triangle and put the work in so that your brain will be more receptive to learning why binomial theorem is so awesome.

Rather than writing out 26 rows of Pascal's triangle, you can use the formula:

n choose k = n! / ( k! * ( n - k )! )

For example, 26 choose 2 = 26! / ( 2! * (26 - 2 )! )

= 26! / ( 2 * 24!)

= 26 * 25 / 2

= 325

So the coefficient of x^24 * y^2 will be 325

Note that n choose ( n - k ) = n choose k, so you only have to do half the work, since this also gave us the coefficient of x^2 * y^24

X^26 + y^26