Greatest binomial coefficient:
In the binomial expansion of (1 + x)n , when n is even, the largest binomial coefficient is provided by nCn/2.
Same as if n be odd, the largest binomial coefficient may be
both being same.
If tr and tr + 1 be the rth and (r + 1)th term in the expansion of (1 + x)n, then
x.
Suppose numerically, tr + 1 be the largest term in the given expansion. Then tr + 1 ³ tr
![718_Greatest binomial coefficient2.png](https://www.expertsmind.com/../CMSImages/718_Greatest binomial coefficient2.png)
......(2)
Now shifting values of n and x in (2), we obtain r £ m + f or r £ m
Where m is a positive integer, f is a fraction so that 0 £ f < 1.
Now if f = 0 then tm + 1 and tm both the parts will be numerically same and bigger while if f ¹ 0, then tm +1 is the greatest term of the binomial expansion.
i.e. to search the largest term (numerically) in the expansion of (1 + x)n.
(i) Search m =
.
(ii) If m is integer, then tm and tm + 1 are same and are biggest term.
(iii) If m is not integer, then t[m] + 1 is the biggest term (where [.] shows the greatest integer function).
Illustration: Calculate the value of the greatest term in the expansion of
.
Solution: Since ![2134_Greatest binomial coefficient6.png](https://www.expertsmind.com/../CMSImages/2134_Greatest binomial coefficient6.png)
tr+1 ≥ tr if only 21 - r ≥r√3
if only![2181_Greatest binomial coefficient7.png](https://www.expertsmind.com/../CMSImages/2181_Greatest binomial coefficient7.png)
Therefore t1 < t2 < t3 < t4 < t5 < t6 < t7 < t8 > t9 > t10
Therefore t8 is the greatest term and its value is ![271_Greatest binomial coefficient8.png](https://www.expertsmind.com/../CMSImages/271_Greatest binomial coefficient8.png)
= ![1180_Greatest binomial coefficient9.png](https://www.expertsmind.com/CMSImages/1180_Greatest%20binomial%20coefficient9.png)
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