Problems in Greatest Binomial Coefficient

Algebra > Binomial Theorem Examples

7. Find the greatest binomial coefficient in the binomial expansion (3x+2y)¹².

Solution:

In the given binomial expansion, the index n is 12, an even number. Therefore from our discussion in Greatest Binomial Coefficients , we see that the greatest binomial coefficient will be ⁿc_{( n/2 + 1)}.

Therefore it is ¹²c_{(12/2 + 1)} = ¹²c₍₆₊₁₎ = ¹²c₇.

8. Find the greatest binomial coefficient in the binomial expansion of
(2x – 4y)¹¹

Solution:

In the given binomial expansion, the index n is 11, an odd number. Therefore from our discussion in Greatest Binomial Coefficients, we see that there are two greatest binomial coefficients. And,they are:

ⁿc_(n+1)/2 and ⁿc_{( n + 3 )/2}.

Applying 11 in n:

¹¹c_(11+1)/2 = ¹¹c₆ and ¹¹c_(11+3)/2 = ¹¹c₇

9. Find the value of

²⁰C₁ + ²⁰C₃ + ²⁰C₅ + ²⁰C₇ + ………….. + ²⁰C₁₉

Solution:

From properties of binomial coefficients , we know that

c₀ + c₂ + c₄ +……….. = c₁ + c₃ + c₅ + …………. = 2^{n – 1}

Since only sum of the odd binomial coefficients is required, we can apply the above formula:

²⁰C₁ + ²⁰C₃ + ²⁰C₅ + ²⁰C₇ + ………….. + ²⁰C₁₉ = 2^{20 – 1}

10. In the binomial expansion (2x² – 3/_x)¹⁵, the term which does not have x is ¹⁵c₁₀ . 3¹⁰ . K, where K is a real number. Find K

Solution.

The term is independent of x ( since it does not have x ). Therefore, we will use general term to find r which is independent of x

T_{r + 1} = ⁿc_r .x^{n – r}. y^r

= ¹⁵c_r ( 2x²) ^{15 – r} ( ^{- 3} / _x) ^r

= ¹⁵c_r (2)^{15 – r} ( - 3 ) ^r (x²) ^{15 – r} (1/_x)^r

= ¹⁵c_r (2)^{15 – r} ( - 3 ) ^r (x^{30 – 2r})(x)^{- r}

= ¹⁵c_r (2)^{15 – r} ( - 3 ) ^r (x ^{30 – 3r})……………………(3)

Since T_{r + 1} is independent of x, equate 30 – 3r to 0

30 – 3r = 0, 3r = 30, r = 10.

Since r = 10, it is the 11^th term that does not have x.

Plug in r = 10 in (3) above to find K.

= ¹⁵c₁₀ (2)^{15 – 10} ( - 3 )¹⁰

= ¹⁵c₁₀ ( 3 )¹⁰ (2)⁵ ……………..(4)

Comparing the value in (4) above with ¹⁵c₁₀ .3¹⁰ . K given in the question, we observe that

K = 2⁵ = 32