Question

The sum of all the coefficient of those terms in the expansion of
${\left( {a + b + c + d} \right)^8}$ which contains b but not c is
A. 6305.
B. ${4^8} - {3^8}$.
C. Number of ways of forming 8-digit numbers using digit 1, 2, 3 each number at least one 3.
D. Number of ways of forming 4-digit numbers using digit 1, 2, 3 each number at least one 3.

Answer 1

Hint: To solve this question, we will use the concept of binomial expansion. We will expand the given term and find out the sum of the coefficient of terms by putting the value of terms as 1.

Complete step-by-step answer:
Given that,
${\left( {a + b + c + d} \right)^8}$. …….. (i)
First let us find the sum of coefficients of the terms in the expansion which does not contain c.
Put a = 1, b = 1, c = 0 and d = 1 in equation (i), we will get
$ \Rightarrow {\left( {a + b + c + d} \right)^8} = {\left( {1 + 1 + 0 + 1} \right)^8}$.
$ \Rightarrow {3^8}$. …………. (ii)
Now, we will find out the sum of coefficients of the terms that do not contain b and c.
Put a = 1, b = 0,c = 0 and d = 1 in equation (i), we will get
$ \Rightarrow {\left( {a + b + c + d} \right)^8} = {\left( {1 + 0 + 0 + 1} \right)^8}$.
$ \Rightarrow {2^8}$. ………. (iii)
Now, we will use the equation (ii) and (iii) to find out the sum of coefficients of terms which contain b but not c.
Subtracting equation (iii) from (ii), we get
The sum of coefficients of terms which contains b but not c = ${3^8} - {2^8} = 6305$.
Hence, we can say that the sum of coefficients of those terms in the expansion of ${\left( {a + b + c + d} \right)^8}$ which contains b but not c is 6305.
Therefore, the correct answer is option (A).

Note: Whenever we ask such types of questions, we should remember the basic points of expansion. First, we have to find out the sum of coefficients of the terms that does not contain c in the expansion. Then we will find out the sum of coefficients of terms that does not contain b and c both in the expansion. After that, by subtracting the sum of coefficient of terms without c from the terms with b and c, we will get the sum of coefficient of terms which contains b but not c.