Let S be the set of the first 11 natural numbers.

Question

Let S be the set of the first 11 natural numbers. Then the number of elements in $A=\{B \subseteq S: n(B) \geqslant 2$ and the product of all elements of $B$ is even $\}$ is $\_\_\_\_$ .

Answer: 1979

Accepted Answer

Let $S={1,2,\ldots ,11}$. The product of all elements of $B$ is even iff $B$ contains at least one even number.

Evens in $S$: $5$ ($2,4,6,8,10$)

Odds in $S$: $6$

Number of subsets containing at least one even:

${2}^{11}-{2}^{6}=2048-64=1984$

(we subtract the subsets made only of odds).

Now enforce $n(B)\ge 2$: subtract the 5 singleton subsets ${2},{4},{6},{8},{10}$.

$1984-5=1979$

$1979$

Solutions · 1