A multiple-choice test contains 10 questions. There are four possible answers fo

Question

A multiple-choice test contains 10 questions. There are four possible answers for each question. (i) In how many ways can a student answer the questions on the test if the student answers every question? (ii) In how many ways can a student answer the questions on the test if the student can leave answers blank? Question 2. Jordan and Joey are getting married! In how many ways can their wedding photographer arrange 6 people in a row from a group of 10 people, where Jordan and Joey are among these 10 people, if (i) Jordan must be in the picture? (ii) both Jordan and Joey must be in the picture? (iii) exactly one of Jordan and Joey is in the picture? Question 3. Show that there are at least six people in California (population: 37 million) with the same three initials who were born on the same day of the year (but not necessarily in the same year). Assume that everyone has three initials and every year has 365 days. Question 4. A coin is flipped 10 times where each flip comes up either heads or tails. How many possible outcomes (i) are there in total? (ii) contain exactly two heads? (iii) contain at most three tails? (iv) contain the same number of heads and tails? Question 5. A circular r-permutation of n people is a seating of r of these n people around a circular table, where seatings are con- sidered to be the same if they can be obtained from each other by rotating the table. (i) Find the number of circular 3-permutations of 5 people. (ii) Find a formula for the number of circular r-permutations of n people. Question 6. (i) What is the coefficient of x^7 in (1 + x)^11? (ii) What is the coefficient of x^101y^99 in the expansion of (2x - 3y)^200.

Explanation / Answer

Q.No1.

A student has 4 choices for each question

Hence he can select any one of the 4 choices. and if he answers all questions,

no of ways = 104=10000 ways

ii) If he can leave some questions blank, then he either answers 10, or 9 or 8 or 7 or......0

So total no of ways = No of ways for answering all 10 + all 9+all8+all 7+....+0

= 104+10C1 (9)4(1) + 10C2 (8)4+10C3 (7)4+...+10C0

= 104+10C1 (9)4+10C2(8)4+....+10C9 + 1

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