Greatest binomial coefficient:

In the binomial expansion of (1 + x)ⁿ , when n is even, the largest binomial coefficient is provided by ⁿC_n/2.

Same as if n be odd, the largest binomial coefficient may be  both being same.

If t_r and t_{r + 1} be the r^th and (r + 1)^th term in the expansion of (1 + x)ⁿ, then

x.

Suppose numerically, t_{r + 1} be the largest term in the given expansion. Then t_{r + 1} ³ t_r

                                  ......(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 t_{m + 1} and t_m both the parts will be numerically same and bigger while if f ¹ 0, then t_{m +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 t_m and t_{m + 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 

                                    t_r+1 ≥ t_r    if only 21 - r ≥r√3

                                    if only

                                    Therefore t₁ < t₂ < t₃ < t₄ < t₅ < t₆ < t₇ < t₈ > t₉ > t₁₀

                                    Therefore t₈ is the greatest term and its value is 

                                                = 

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