Binomial Theorem : CAT Quantitative Aptitude Questions

Binomial Coefficients
The number ⁿC_r are also called binomial coefficients. The reason for this is that they appear as coefficients of powers of x and y in the expansion of (x+y)ⁿ . The binomial theorem states that for all real numbers x and y for all positive integers n

Note: There are (n + 1) terms in the above expansion of (x+y)ⁿ . The (r + 1)^th term is called the General Term of the Expansion and is denoted by T_{r + 1}

General Term

The General Term,

Important Corollary

If we write ‘– y’ in place of yin (i), we obtain:

 
Note: The terms in the expansion of (x+y)ⁿ  and (x-y)ⁿ are numerically the same except that in (x-y)ⁿ they are alternatively positive and negative, depending on n being odd or even.

General Term

The General Term,

Example:

How many different subsets are there of a set consisting of n elements?

Solution:

There are nCr different subsets consisting of k elements for k = 0, 1, 2, …,n.
The total number of subsets isⁿC₀ + ⁿC₁ + … + ⁿC_n = (1 + 1)ⁿ = 2_ⁿ

The same result could have also been obtained from the fundamental principle of counting. A subset is determined by the elements it contains. For each of the n elements in the original set there are two possibilities: it may or may not be in the subset.

Example:
If the sum of the fifth and the sixth terms is zero in the binomial expansion of (a - b)ⁿ , n ≤ 5, then the value of a/b is:



Solution:
Since there are (n + 1) terms in the binomial expansion of (a - b)ⁿ , fifth and sixth terms will be present only in the expansion of (a - b)⁵ .

therefore Fifth term in the binomial expansion of

Similarly, Sixth term

Now, the problem states that the sum of the fifth and the sixth terms is zero,

Hence, option [4] is the correct answer.