(Solved): answer all please (question 7 - 11 ) 7. A rectangle is partitioned into 5 regions as shown. Each reg ...

7. A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color-red. orange, yellow, blue, or green-so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible? (A) 120 (B) 270 (C) 360 (D) 540 (E) 720 8. The infinite product evalistet to a real number. What is that number? \( \sqrt[3]{10} \cdot \sqrt[3]{10}+\sqrt[3]{\sqrt[3]{\sqrt[3]{10}} \cdots} \) Cvaliatet to at real number. What is that number? \( (\mathrm{A}) \sqrt{\operatorname{T0}} \) (B) \( \sqrt[3]{100} \) (C) \( \sqrt[4]{1000} \) (D) 10 (E) \( 10 \sqrt[3]{10} \) 9. On Halloween 31 children walked into the principal's office asking for candy. They can be classified inte three types: Sorne always lie; some always tell the truth; and some alternately lie and tell the trath. The alfematers arbitrarily choose their first response, either a lie or the truth, but each subsequent stafement has the opposite truth value from its predecessor The principal asked everyone the same three quetions in this order "Are you a truth-teller? The prineipal gave a piece of eandy to each of the 22 children who' anwwered yes. "Are you an alternater?" The principal gave a piece of candy to each of the 15 childen who answered "Are you a liar?' The principal gave a piece of candy to cach of the 9 children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth? (A) 7 (B) 12 (C) 21 (D) 27 (E) 31 10. How many ways are there to split the integers 1 through 14 into 7 pairs so that in each pair the greater number is at least 2 times the lesser number? (A) 108 (B) 120 (C) 126 (D) 132 (E) 144 1. What is the product of all real numbers \( x \) such that the distance on the number line between log. \( x \) atid loge 9 is twice the distance on the number line between \( \log _{6} 10 \) and \( 1 ? \) (A) 10 (B) 18 (C) 25 (D) 36 (E) 81