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 $=43$