1972 AHSME Problems/Problem 4

Problem 4

The number of solutions to $\{1,~2\}\subseteq~X~\subseteq~\{1,~2,~3,~4,~5\}$, where $X$ is a subset of $\{1,~2,~3,~4,~5\}$ is

$\textbf{(A) }2\qquad \textbf{(B) }4\qquad \textbf{(C) }6\qquad \textbf{(D) }8\qquad  \textbf{(E) }\text{None of these}$

Solution

$X$ has to contain $\{1,~2\}$, so only $\{3,~4,~5\}$ matters. There are two choices for the elements; the element is either in $X$ or outside of $X$. With this combinatorics in mind, the answer is simply $2^3=\boxed{\textbf{(D) }8}.$ ~lopkiloinm

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