Now You Try

<p>Each player is dealt six cards from a standard deck. In how many ways could a hand contain at least two queens?</p>

<p>A set of 12 distinct wooden blocks has three that are red, five that are blue, and four that are yellow. The blocks are labeled R1, R2, R3, B1, ..., B5, Y1,..., Y4, where the letter matches the colour. Consider the following subsets of U, the set of all possible subset of two blocks.</p>
<ul>
<li><code class='latex inline'>A</code>: all subsets of two red blocks.</li>
<li><code class='latex inline'>B</code>: all subsets with one red and one yellow block.</li>
<li><code class='latex inline'>C</code>: all subsets with two blocks the same colour.</li>
</ul>
<p>a. Find <code class='latex inline'>n(U), n(A), n(B), n(C)</code>.</p><p>b. How many subsets have two blocks of different colours?</p>

<p>Identify whether the
following situations involve permutations, combinations, or both. Justify your choice.</p><p>Naming 3 people from among
15 contestants to win 3 different prizes</p>

<p>A judging panel will have 6 members chosen from 8 teachers and 10 students. There must
be at least 3 students on the panel. In how many ways could there be</p><p>a) 3 students on the panel?</p><p>b) 4 students on the panel?</p><p>c) 5 students on the panel?</p><p>d) least 3 students on the panel?</p><img src="/qimages/45456" />

<p>How many sequences of length 4 can be constructed using the digits <code class='latex inline'>\{1, 2, 3, 4, ..., 9\}</code> if two of the terms are even and two are odd?</p>

<p>Tonya has the following toppings available for her sandwich: lettuce, tomatoes, onions, olives, sprouts, peppers, mustard, and shredded cheese. She can use up to three toppings. How many different sandwiches can Tonya make?</p><img src="/qimages/45760" />

<p>If a set has 12 elements, how many subsets can be formed?</p>

<p>In cribbage, each player is dealt six cards from a standard deck. In how many ways could a hand contain</p><p>at least two hearts and at least two spades?</p>

<p>Six different dice are rolled. What fraction of all the possible outcomes have two 2s, two 4s, and two 6s.</p>

<p>How many of the anagrams of the word Mississauga are palindromes using up to 5 letters from it.</p>

<p>A subset of 3 blocks is selected from a population of six blocks, of which three are red. A student is trying to count the number of subsets with at least two red blocks. She reasons that she can select the two red blocks in C(3, 2) ways and then the remaining block in <code class='latex inline'>C(4, 1)</code> ways. Using the product rule, the student calculates that there are <code class='latex inline'>C(3, 2) \times C(4, 1) = 12</code> subsets with at least two red blocks. Label the blocks <code class='latex inline'>R1, R2, R3, A, B, C</code>, where the first three blocks are red.</p><p>a. List all subsets that have at least two red blocks and count them directly.</p><p>b. Explain why the student got the wrong answer.</p>

<p>You receive requests to connect with people every day on your social media account.</p><p>If you have 15 requests to be “friends” with people, in how many ways could you
respond by either accepting or rejecting each request?</p>

<p>A store has 30 female employees and 20 male employees. The manager wants to select 10 employees to help choose a new uniform.</p><p>a) Identify the population.</p><p>b) Describe how the manager can choose a stratified random sample.</p>

<p>Suppose a sequence of length 4 is formed by choosing four digits from the set<code class='latex inline'>\{0, 1, 2, ...., 9\}</code>. The 4 digit number must be a unique 4 digit number.</p><p>a. For any subset of size 4 selected from the above set of digits, how many different sequences of length 4 can be formed?</p><p>b. By counting the sequences in two ways, explain why <code class='latex inline'>C(10, 4) = \frac{P(10 ,4)}{4!}</code>.</p>

<p>To learn how students feel about a proposed dress code, the principal decides to survey a sample of 60 students from the school population of 1200 students.</p><p>a. How many different samples can be selected?</p><p>b. If there are 300 students in each grade from 9 to 12, how many samples can be selected that have 15 students in each grade?</p><p>c. How many samples can be selected that have 60 grade 12 students?</p>

<p>In how many ways could a group of 10 people form a committee with at least 8 people on it?</p>

<p>A cabin has two rooms with three single beds each, one room with four single beds, and one room with two single beds. </p><p>Six girls and six boys are assigned to rooms with people of the same gender. In how many ways can the rooms be assigned?</p>

<p>On a crown and anchor wheel, a crown, an anchor. and the four suits from a deck of cards are displayed in slots around the wheel.</p><p>The following restrictions are in place when all three symbols are different:</p>
<ul>
<li>A crown and an anchor do not occur
together (e.g., whiz. cannot occur). </li>
<li> Three different suits do not occur together . </li>
<li>If a crown occurs with two different suits, an anchor may not also occur
with the same two suits, and vice versa</li>
</ul>
<p>Calculate the number of slots with three different symbols. Use your calculations to verify the total number of slots on the wheel.</p>

<p>How many palindromes are there for the word Mississippi using up to 5 letters from it.</p>

<p>A board of directors needs to assign a chair, vice-chair, treasurer, secretary, and
communications officer. There are four women and six men on the board. There w111' be two women and three men on the executive. In how many ways could this be done?</p>

<p>You have three $5 bills, a $10 bill, and two $20 bills. How many different sums of money can you make?</p>

<p>On a crown and anchor wheel, a crown, an anchor, and the four suits from a deck of cards are displayed in slots around the wheel.</p><p>Determine the number of slots with two-of-a-kind.</p>

<p>In a lottery, numbered balls are selected from 49 balls numbered 1, 2, ..., 49. For the following questions, express your answers in terms of <code class='latex inline'>C(n, r)</code> for various choices of <code class='latex inline'>n</code> and <code class='latex inline'>r</code>.</p><p>a. How many possible subsets of six balls are there?</p><p>b. How many of these subsets contain ball 49?</p>

<p>There are 10 points in a plane. No three points are collinear. How many convex polygons can be drawn using these points as vertices?</p>

<p> A subset of five numbers is chosen from the set <code class='latex inline'>\{1, 2, ...., 10\}</code>.</p><p>a. How many subsets can be selected?</p><p>b. How many of these subsets contain only numbers less than or equal to 7?</p><p>c. How many of the subsets contain two even and three odd numbers?</p><p>d. How many of the subsets contain at least two even numbers?</p><p>e. How many of the subsets contain the number 10?</p><p>f. How many of the subsets contain 9 or 10?</p>

<p>A telemarketer will call 12 people from a list of 20 men and 25 women. In how many ways could he select</p><p><em>a)</em> 12 men or 12 women?</p><p><em>b)</em> 6 men and 6 women?</p>

<p>Points are drawn on a circle.</p><img src="/qimages/5288" /><p>a) If there are three points, how many line segments can be drawn joining any two points?</p><p>b) What if there are four points? five points? six points?</p><p>c) If there are <code class='latex inline'>n</code> points, how many line segments can be drawn joining any two points?</p><p>d) How many points are needed in order to have at least <code class='latex inline'>1000</code> line segments?</p>

<p>How many committees of size 5 can be selected from 11 people - five men and six women- if</p><p>a. there are no restrictions?</p><p>b. the committee has three women and two men?</p><p>c. the committee must contain at least one man and one woman?</p><p>d. Tom and Enzo refuse to serve on the same committee?</p>

<p>In cribbage, each player is dealt six cards from a standard deck. In how many ways could a hand contain</p><p>more than three red cards?</p>

<p>On a crown and anchor wheel, a crown, an anchor. and the four suits from a deck of cards are displayed
in slots around the wheel.</p><p>Each three-of—a-kind (e.g., crown, crown, crown) occurs twice. Calculate the number of slots with three-of—a-kind.</p>

<p>A direct road needs to be built between each pair of the six towns arranged in a normal hexagon. How many roads need to be built?</p>

<p>How many arrangements can be formed from <code class='latex inline'>a A</code>s, <code class='latex inline'>b B</code>s and <code class='latex inline'>c C</code>s if no two of the As are consecutive?</p>

<p>Identify whether the
following situations involve permutations, combinations, or both. Justify your choice.</p><p>Forming a committee of 5 people from a group of 12 people</p>

<p>How many different sums of money can be made from a $5 bill, a $10 bill, a $20 bill, and
a $50 bill?</p>

<p>Suppose a sequence of length <code class='latex inline'>4</code> is formed by choosing four digits from the set<code class='latex inline'>\{0, 1, 2, ...., 9\}</code>. The 4 digit number must be a unique 4 digit number.</p>
<ul>
<li>How many such sequences can be formed if no term in the sequences can be repeated?</li>
</ul>

<p>In a lottery, numbered balls are selected from 49 balls numbered 1, 2, ..., 49. For the following questions, express your answers in terms of <code class='latex inline'>C(n, r)</code> for various choices of <code class='latex inline'>n</code> and <code class='latex inline'>r</code>.</p><p>a. How many of these subsets contain only even-numbered balls?</p><p>b. How many of these subsets contain three even numbered and three odd numbered balls?</p>

<p> A subset of size 3 is formed by selecting three letters from the set <code class='latex inline'>\{A, B, C, D, E, F, G\}</code>. What fraction of the possible subsets</p><p>contain exactly one vowel?</p>

<p>Six different dice are rolled. What fraction of all the possible outcomes have at least one repeated value?</p>

<p>You can factor the number 210 into prime factors as <code class='latex inline'>2 \times 3 \times 5 \times7</code>.The products of prime factors form divisors (e.g., <code class='latex inline'>2 \times 3 = 6</code>). Determine the total number of divisors of 210.</p>

<p>Rohan is shopping for new pants. Six' different styles are available. How many different purchases could Rohan make?</p>

<p>Identify whether the
following situations involve permutations, combinations, or both. Justify your choice.</p><p>Choosing 4 men and 4 women to be on a basketball team from among 6 men and 6 women, and assembling the athletes for a team photo</p>

<p> Seven friends have initials O, P, Q, R, S, T, and U. Each must have a telephone conversation with friends whose initials are within two letters of their own. Use a diagram and a numeric representation to determine how many telephone conversations will occur.</p>

<p>Identify whether the
following situations involve permutations, combinations, or both. Justify your choice.</p><p>Choosing a president, a vice president, and a treasurer from a committee of 12 members</p>