Which of the following terms appear in the expansion of (x + y)10? The letter a in each term represents a real constant.
ax³y⁷, ay⁵, ax⁵, ax⁵y

Solution:

The binomial expansion (a + b)ⁿ = aⁿ + na^{n - 1} b + n(n - 1) a^{n - 2}b² / 1.2 + n(n - 1)(n - 2)a^{n - 3}b³ / 1.2.3 + ……..+ nab^{n - 1} + bⁿ

Looking at the terms above one can observe that the sum of the powers of a and b add to give the sum of n.

The first term is aⁿ(aⁿb⁰) and the sum of powers of a and b is n + 0 = n 

The second term na^{n - 1}b has the sum of powers of a and b as n - 1 + 1 = n

The third term is n(n - 1)a^{n - 2}b² / 1.2 and the sum of powers of a and b is n - 2 + 2 = n

This will be observed for the higher terms also. Therefore in the expansion of (x + y)¹⁰. The term which will appear in the expansion: will be:

ax³y⁷    (The sum of the powers of x and y is 3 + 7 = 10)

Which of the following terms appear in the expansion of (x + y)¹⁰? The letter a in each term represents a real constant.

Summary:

The term that appears in the expansion of (x + y)¹⁰ is ax³y⁷ ..