Click to Chat

1800-1023-196

+91-120-4616500

CART 0

• 0

MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: Rs.

There are no items in this cart.
Continue Shopping

```Binomial Coefficients

We know that,

(a + x)n = nC0 an + nC1 an-1 x + nC2 an-2 x2 +...+ nCn xn                ... (i)

Let us write equation (i) in particular form by putting a = 1. Now it becomes,

(1 + x)n = nC0 + nC1 x + nC2 x2 +...+ nCr xr +...+ nCn xn                   ... (ii)

Coefficients attached with different powers of x are called Binomial Coefficients.

Now, again put a = 1 in the expression

(a + x)n = an + n an-1 x + (n(n-1))/2! an-2 x2 + (n(n-1)(n-2))/3! an-3 x3

+...+ (n(n-1)(n-2)....(n-r+1))/r! an-r xr +......+ xn

we get,

(1+x)n = 1 + nx + (n(n-1))/2!  x2 +...+ (n(n-1)(n-2)......(n-r+1))/r!  xr +...+ xn            ...(iii)

If we compare coefficient of xr in expansions (ii) and (iii), we have,

nCr = (n(n-1)(n-2)......(n-r+1))/r! = n!/r!(n-r)!

What does it mean when we compare two expansions?

Since expansion (ii) is valid for any value of x, we can replace x by 1/x , (x ≠ 0) in it, we get,

(1 + (1/x))n  =  nC0 + nC1 1/x + nC2  1/x2 +...+ nCr  1/xr +...+ nCn  1/xn

Multiplying both sides by xn, it becomes,

(1+x)n = nC0 xn + nC1 xn-1 +...+ nCr xn-r +...+ nCn                        ... (iv)

Compare expansion (ii) and (iv). It will give us, that

nCr = nCn-r, this is a beautiful result, which says that in the expansion of the equations of the like of (ii) the rth coefficient from the beginning is equal to the rth coefficient from the end.

AskIITians offers a novel way of teaching where you can prepare for IIT JEE, AIEEE and other engineering examinations for free by sitting at home. You can visit the website askIITians.com to read the study material pertaining to your preparation. Be a part of our online tests and AQAD (A Question A Day) for free and be a winner.

To read more, Buy study materials of Binomial Theorem comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.
```

### Course Features

• 731 Video Lectures
• Revision Notes
• Previous Year Papers
• Mind Map
• Study Planner
• NCERT Solutions
• Discussion Forum
• Test paper with Video Solution