Permutation Combination Question 4
Question 4 - 25 January - Shift 1
Let $S={1,2,3,5,7,10,11}$. The number of nonempty subsets of $S$ that have the sum of all elements a multiple of 3 , is _______
Show Answer
Answer: 43
Solution:
Formula: Sum of numbers
Elements of the type $3 k=3$
Elements of the type $3 k+1=1,7,9$
Elements of the type $3 k+2=2,5,11$
Subsets containing one element $S_1=1$
Subsets containing two elements
$S_2={ }^{3} C_1 \times{ }^{3} C_1=9$
Subsets containing three elements
$S_3={ }^{3} C_1 \times{ }^{3} C_1+1+1=11$
Subsets containing four elements
$S_4={ }^{3} C_3+{ }^{3} C_3+{ }^{3} C_2 \times{ }^{3} C_2=11$
Subsets containing five elements
$S_5={ }^{3} C_2 \times{ }^{3} C_2 \times 1=9$
Subsets containing six elements $S_6=1$
Subsets containing seven elements $S_7=1$
$\Rightarrow$ sum $=43$
