How to tell the difference between permutation and combination
Permutations and Combinations
Combinations and Permutations Word Problems
01 Luka is making lemonade to sell at a school fundraiser. His recipe requires 4 times as much water as sugar and twice as much sugar as lemon juice. He uses 3 cups of lemon juice. How many cups of water does he need?
Luka正在制作柠檬水，准备在学校的募捐活动中出售，他的配方中水是糖的4倍，并且糖是柠檬汁的两倍。他用了3杯柠檬汁，问他需要多少杯水？
(A) 6 (B) 8 (C) 12 (D) 18 (E) 24
正确答案：（E)24
02 Four friends do yardwork for their neighbors over the weekend, earning $15, $20, $25, and $40, respectively. They decide to split their earnings equally among themselves. In total how much will the friend who earned $40 give to the others?
四个朋友在周末为他们的朋友整理庭院，分别赚了15美元，20美元，25美元和40美元。他们决定平均分配收入。问赚了40美元的朋友总共会给其他人多少钱？
(A) $5 (B) $10 (C) $15 (D) $20 (E) $25
正确答案：（C)15
03 Carrie has a rectangular garden that measures 6 feet by 8 feet. She plants the entire garden with strawberry plants. Carrie is able to plant 4 strawberry plants per square foot, and she harvests an average of 10 strawberries per plant. How many strawberries can she expect to harvest?
Carrie有一个长方形花园，长6英尺，宽8英尺。她把整个花园都种上了草莓。Carrie能在每平方英尺种4株草莓，并且她平均每株收获10个草莓，问她预计能收获多少个草莓？
(A) 560 (B) 960 (C) 1120 (D) 1920 (E) 3840
正确答案：（D)1920
04 Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
三个不断变大的六边形如图所示。假设点阵排列的规则继续，每接下来的一个六边形都会多一个层点。问下一个六边形里有多少点？
(A) 35 (B) 37 (C) 39 (D) 43 (E) 49
正确答案：（B)37
05 Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of 5 cups. What percent of the total capacity of the pitcher did each cup receive?
一个壶的四分之三装有菠萝汁。将等量的果汁倒入5个杯子中的每一个后，壶空了。问每个杯子里的果汁是壶的总容量的百分之多少(A) 5 (B) 10 (C) 15 (D) 20 (E) 25
正确答案：（C)15
06 Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in frond of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
Aaron，Darren，Karen，Maren,和Sharon坐上了一列有五节车厢的小火车，每节车厢可坐一人。Maren坐在最后一节车厢。Aaron坐在Sharon的背后。Darren坐在Aaron前面的一节车厢里。Karen和Darren中间至少隔着一个人。问谁坐在正中间的车厢里？(A) Aaron (B) Darren (C) Karen (D) Maren (E) Sharo
正确答案：（A)Aaron
07 How many integers between 2020 and 2400 have four distinct digits arranged in increasing order? (For example, 2357 is one such integer.)
在2020和2400之间，四位数字互不相同，并且是按照递增顺序排列的整数有多少个？（例如，2357就是这样一个整数）
(A) 9 (B) 10 (C) 15 (D) 20 (E) 28
正确答案：（C)15
08 Ricardo has 2020 coins, some of which are pennies (1-cent coins) and the rest of which are nickels (5-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least possible amounts of money that Ricardo can have?
Ricardo有2020枚硬币，其中一些是1美分硬币，其余的是5美分硬币。他至少有一枚1美分硬币和一枚5美分硬币。Ricardo能拥有的最大可能的金额和最小可能的金额之间的差值是多少美分？
(A) 8062 (B) 8068 (C) 8072 (D) 8076 (E) 8082
正确答案：（C) 8072
09 Akash’s birthday cake is in the form of a 4×4×4 inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into 64 smaller cubes, each measuring 1×1×1 inch, as shown below. How many of the small pieces will have icing on exactly two sides?
Akash的生日蛋糕是4×4×4英寸的立方体。蛋糕的顶部和四个侧面都有糖霜，底部没有糖霜。如图所示，假设将蛋糕切成64个小方块，每个小方块的尺寸是1×1×1英寸。问恰好两面有糖霜的小方块有多少个？
(A) 12 (B) 16 (C) 18 (D) 20 (E) 24
正确答案：（D) 20
10 Zara has a collection of 4 marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
Zara收集了4个弹珠：Aggie,Bumblebee,Steelie和Tiger。她想把它们摆成一排放在架子上展示，但又不想让Steelie和Tiger挨在一起。问她有多少种排法？
(A) 6 (B) 8 (C) 12 (D) 18 (E) 24
正确答案：（C) 12
11 After school, Maya and Naomi headed to the beach, 6 miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi’s and Maya’s average speeds?
放学后，Maya和Naomi前往6英里外的海滩。Maya决定骑自行车，而Naomi则乘坐公交车。上图展示了他们的旅程，包括时间和行程。问Naomi和Maya的平均速度的差值是多少英里每小时？
(A) 6 (B) 12 (C) 18 (D) 20 (E) 24
正确答案：（E) 24 algebra: motion, averages speed, and graphs
12 For positive integer n, the factorial notation n! represents the product of the integers form n to 1. (For example, 6! 6 5 4 3 2 1 = ⋅⋅⋅⋅⋅ ) What value of N satisfies the following equation?
对于正整数n,阶乘符号n！表示从n到1的整数的乘积。（例如：6！= 6×5×4×3×2×1。）问满足下面方程的N的值是多少？
5！× 9！=12×N！
(A) 10 (B) 11 (C) 12 (D) 13 (E) 14
正确答案：（A) 10 algebra: factorials and rewriting expressions
13 Jamal has a drawer containing 6 green socks, 18 purple socks, and 12 orange socks. After adding more purple socks, Jamal noticed that there is now a 60% chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
Jamal的一个抽屉里面有6只绿色袜子，18只紫色袜子和12只橙色袜子。在添加更多紫色袜子后，Jamal发现现在抽屉里随机抽取一只袜子是紫色的概率是60%。问Jamal一共添加了多少只紫色袜子？
(A) 6 (B) 9 (C) 12 (D) 18 (E) 24
正确答案：（B) 9 arithmetic: ratios and probability
14 There are 20 cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all 20 cities?
牛顿郡有20个城市。它们的人口数量如柱状图所示。所有城市的平均人口数量以水平虚线表示。问以下哪个选项最接近所有20个城市的人口总数？
(A) 65,000 (B) 75,000 (C) 85,000 (D) 95,000 (E) 105,000
正确答案：（D) 95000 arithmetic: graphs, averages, and estimation
15 Suppose 15% of x equals 20% of y. What percentage of x is y?
假设x的15%与y的20%相等，y是x的百分之多少？
(A) 5 (B) 35 (C) 75 (D) 13313 (E) 300
正确答案：（C) 75 algebra: percents
16 Each of the points A, B, C, D, E, and F in the figure below represents a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is 47. What is the digit represented by B?
图中的点A,B,C,D,E和F各自代表从1至6的不同数字。如图所示的五条线中每条线都经过其中的一些点。将每条线上的数字相加可以生产五个和，每条线一个。五个和的总和是47。问B所代表的数字是多少？
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
正确答案：（E) 5 algebra: sums and equations
17 How many factors of 2020 have more than 3 factors? (As an example, 12 has 6 factors, namely 1, 2, 3, 4, 6, and 12.)在2020的约数中有多少个数有3个以上的约数？（举个例子，12有6个约数，即1,2,3,4,6和12。）
(A) 6 (B) 7 (C) 8 (D) 9 (E) 10
正确答案：（B) 7 number theory: factors and number of factors
18 Rectangle ABCD is inscribed in a semicircle with diameter FE, as shown in the figure. Let DA=16, and let FD=AE=9. What is the area of ABCD?
如图所示，长方形ABCD内接在直径为FE的半圆中。已知DA=16，并且FD=AE=9。问ABCD的面积是多少？
(A) 240 (B) 248 (C) 256 (D) 264 (E) 272
正确答案：（A) 240 geometry: circles, rectangles, and Pythagorean theorem
19 A number is called flippy if its digits alternate between two distinct digits. For example, 2020 and 37373 are flippy, but 3883 and 123123 are not. How many five-digit flippy numbers are divisible by 15?
一个数如果由两个交替出现的不同数字组成，就成为交替数。例如2020和37373是交替数，而3883和123123不是。问有多少个五位交替数能被15整除？
(A) 3 (B) 4 (C) 5 (D) 6 (E) 8
正确答案：（B) 4 number theory: divisible and possible digits
20 A scientist walking through a forest recorded as integers the heights of 5 trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
一位科学家在森林中行走，用整数将一排5棵树的高度记录下来。她观察到每棵树的高度都是它右边那棵树高度的两倍或一半。不幸的是，雨水滴落在她的笔记本上，因而丢失了一些数据。她的笔记本如上图所示，横线表示丢失的数。这位科学家能够根据她的观察重建丢失的数据。问这些树木的平均高度是多少米？
(A) 22.2 (B) 24.2 (C) 33.2 (D) 35.2 (E) 37.2
正确答案：（B) 24.2 arithmetic and logic: averages and remainders
21 A game board consists of 64 squares that alternate in color between black and white. The figure below shows square P in the bottom row and square Q in the top row. A market is placed at P. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7-step paths are there from P to Q? (The figure shows a sample path.)
一块游戏板由64个黑白相间的方格组成。如图所示，方格P在最下面一行，方格Q在最上面一行。在P处有一枚棋子，一步是指将棋子从一个方格移到上面一行的相邻的白色方格。问从P到Q有多少条7步路径？（图中给出了一条路径示例）
(A) 28 (B) 30 (C) 32 (D) 33 (E) 35
正确答案：（A) 28 counting and probability: counting paths and patterns
22 When a positive integer N is fed into a machine, the output is a number calculated according to the rule shown below.
当一个正整数被输入机器时，输出的是一个按上图规则计算的结果。例如，从输入N=7开始，机器将输出3*7+1=22。接着若将结果不断向机器中输入五次，最后输出的结果是26。
For example, starting with an input of N=7, the machine will output 3 7 1 22 ⋅ + = . Then if the output is repeatedly inserted into the machine five more times, the final output is 26.
当同样的6个步骤被应用于不同的起始值N时，最后输出的结果为1，那么所有这样的整数的总和是多少？
When the same 6-step process is applied to a different starting value of N, the final output is 1. What is the sum of all such integers N?
当同样的6个步骤被应用于不同的起始值N时，最后输出的结果为1，那么所有这样的整数的总和是多少？
(A) 73 (B) 74 (C) 75 (D) 82 (E) 83
正确答案：（E) 83 number theory: sequences and recursion
23 Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?
五个不同的奖项将颁发给三名学生。每名学生将至少获得一个奖项。问颁发奖项的不同方式共有多少种？
(A) 120 (B) 150 (C) 180 (D) 210 (E) 240
正确答案：（B) 150 counting and probability: cases and logic
24 A large square region is paved with n2 gray square tiles, each measuring s inched on a side. A border d inches wide surrounds each tile. The figure below shows the case for n=3. When n=24, the 576 gray tiles cover 64% of the area of the large square region. What is the ratio ds for this larger value of n?
一个大的正方形区域铺上了n2块灰色的方砖，每块方砖的边长为s英寸。每块瓷砖周围都有一个d英寸宽的边。上图显示的是n=3的情况。当n=24时，576块灰色瓷砖覆盖了大正方形区域64%的面积。对于这个较大的n值，比例d/S是多少？
（A）9/25 （B）7/16 （C）9/16 （D）1/4 （E）6/25
正确答案：（E) 6/25 geometry and algebra: areas, percentages, and ratios
25 Rectangles R1 and R2, and squares S1, S2 and S3, shown below, combine to from a rectangle that is 3322 units wide and 2020 units high. What is the side length of S2 in units?
如图所示，长方形R1和R2，正方形S1，S2和S3，组合成一个宽3322单位，高2020单位的长方形。问S2的边长是多少单位？
(A) 651 (B) 655 (C) 656 (D) 662 (E) 666
正确答案：（A) 651 geometry and algebra: lengths and equations
