Question

Let $A=\left\{x|x\le 9,x\in N\right\}$. Let $B=\left\{a,b,c\right\}$ be the subset of $A$ where $\left(a+b+c\right)$ is a multiple of $3$. What is the largest possible number of subsets like $B$?

• A. $12$
• B. $21$
• C. $27$
• D. $30$

Answer 1

The correct option is D

$30$

Explanation for the correct Option(s)

Find the largest possible number of subsets like $B$

Given,

$A=\left\{x|x\le 9,x\in N\right\}=\left\{1,2,3,4,5,6,7,8,9\right\}$

$B=\left\{a,b,c\right\}$

The total possible multiple of $3$ can be expressed as

$\{3,6,9,12,15,18,21,24,27\}$

Since $3$ and $27$ cannot be expressed as such that $\left(a+b+c\right)$ is a multiple of $3$ , $3$ and $27$are not possible.

Remaining possible multiple of $3$ are $6,9,12,15,18,21,24$, this can be expressed as such that $\left(a+b+c\right)$ as shown below:

$$\begin{gathered} 6\to 1+2+3 \\ 9\to 2+3+4,5+3+1,6+2+1 \\ 12\to 9+2+1,8+3+1,7+1+4,7+2+3,6+4+2,6+5+1,5+4+3 \\ 15\to 9+4+2,9+5+1,8+6+1,8+5+2,8+4+3,7+6+2,7+5+3,6+5+4 \\ 18\to 9+8+1,9+7+2,9+6+3,9+5+4,8+7+3,8+6+4,7+6+5. \\ 21\to 9+8+4,9+7+5,8+7+6 \\ 24\to 9+8+7 \end{gathered}$$

From the above subsets, the total largest possible number of subsets are $30$.

Hence, the correct answer is option D.