$A$ bag contains $12$ two-rupee coins,$7$ one-rupee coins,and $4$ fifty-paise coins. If three coins are selected at random,the probability that the sum of the values of the three coins is not an integral multiple of a rupee is:

• A
  $\frac{4({ }^{12} C_2 \cdot{ }^7 C_2+{ }^{12} C_1 \cdot{ }^7 C_1+{ }^7 C_2)+3({ }^{12} C_1+{ }^7 C_1)}{{ }^{23} C_3}$
• B
  $\frac{4({ }^{12} C_1 \cdot{ }^7 C_1+{ }^{12} C_2+{ }^7 C_2)+{ }^4 C_3}{{ }^{23} C_3}$
• C
  $\frac{4({ }^{12} C_2 \cdot{ }^7 C_1+{ }^{12} C_1 \cdot{ }^7 C_2)+3({ }^{12} C_1 \cdot{ }^7 C_2)}{{ }^{23} C_3}$
• D
  $\frac{4({ }^{12} C_3+{ }^7 C_3)+3({ }^{12} C_1+{ }^7 C_1)}{{ }^{23} C_3}$