The values of the coefficients are given in the following table for several The table is called Pascal's triangle. values of n. The coefficients in each row in the table may be calculated from those in the preceding row by the following rule: In any row add to a coefficient the following coefficient and place the sum below the latter. As an application, consider the Theorem. The total number of combinations of n things taken one at a time, two at a time, and so on up to n at a time is 2o – 1. In the binomial expansion for (x + a)" let x = a = 1. 1. What is the middle coefficient in the expansion of (a + b)14? Using Pascal's triangle write the coefficients of the expansion (a + b)11. 2. Plot the terms of the expansion (+)12 as ordinates at equal distances along the x-axis. 3. How many compounds consisting of two elements could be made from eighty-three chemical elements? How many consisting of three elements? 4. From eight men, in how many ways can a selection of four men be made (a) which includes two specified men? (b) which excludes two specified men? 5. How many symbols would be available for a cipher if each symbol is an arrangement of the letters a, b, in a group of five. Thus, A B = aaaab, etc. = ааааа, 6. Twelve competitors run a race for three prizes. In how many ways is it possible that the prizes may be given? 7. In how many ways can a baseball nine be arranged if each of the nine players is capable of playing any position? If A must pitch and B, C, D, play in the outfield? If A or B must pitch, B or C catch, and D, E, F, play on the bases? 8. How many dominoes are there in a set from double blank to double six? 9. How many melodies consisting of four notes of equal duration can be formed from the eight tones of the major scale? From the thirteen tones of the chromatic scale? 10. A Yale lock contains 5 cylinders, each capable of being placed in 10 distinct positions, and opens for a particular arrangement of the cylinders. How many locks of this kind can be made so that no two shall have the same key? 11. The combination of a safe consists of figures and letters arranged on three wheels, one bearing the numbers 0 to 9 inclusive, another the letters A to M inclusive, and the third the letters N to Z inclusive. If the safe opens for but one of these arrangements, how many different combinations can be used? 12. How many arrangements of all the letters of the word Columbia (a) begin with a vowel? (b) begin with a consonant and end with a vowel? (c) have the vowels together? 13. Show that the number of ways in which n things can be arranged in a circle is (n − 1)! In how many ways can six persons be arranged in a line? In a circle? 14. Show that if of n things a are alike, b others are alike, c others are alike, etc., the number of distinct permutations, taken all at a time, is n! a! b! c! 15. How many distinct arrangements can be made of all the letters of the word Mississippi? International? 16. How many signals can be made by arranging 2 white flags, 3 red, and 1 blue in a row? 17. Prove that Cr+nCr-1 = n+1Cr. Compare this with the rule fo finding numbers in Pascal's triangle. 18. Prove that 2nC2+r+1 = 2nC2+r (7 n-r n+r+ a result which will be used later. 19. How many different sums can be formed with a penny, a nickle, a dime, a quarter, a half dollar, and a dollar? 20. A set of weights consists of 1, 2, 4, 8, and 16 ounce weights. How many different amounts can be weighed? 21. If three coins are tossed in how many ways can they fall? Solve the problem for 4 coins. 22. If two dice are thrown in how many ways can they fall? 23. In how many ways can the hands of whist be dealt? 24. How many four-figure numbers can be formed with the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, which are (a) divisible by two? (b) divisible by five? 135. Probability. On one of the faces of a cube is placed the letter A, on two of the faces the letter B, and on the remaining three faces the letter C. If the cube is thrown the total number of ways the cube can fall is six, all of which we will assume are equally likely to occur. The number of ways that the letters, A, B, C, can turn up are respectively one, two, and three. In a great number of trials the letter A would turn up approximately in th of the total number of trials, the letter B in ths and the letter C in ths. This does not mean that in every set of six trials A turns up once, B twice, C three times, but that in the long run, as the number of trials is increased, the frequency with which A, B, C turn up approximates to , 2, of the total number of trials. DEFINITION. The ratio of the number of ways in which a particular form of an event may occur to the total number of ways in which the event can occur (all assumed equally likely) is said to be the probability of the particular event. Theorem 1. If the probability that an event will happen is p and the probability that it will not happen is q, then q = 1 - p. Let T denote the number of ways the event can happen, F the number of ways in which the favorable form of the event can happen, and U the number of ways in which the favorable form of the event cannot happen. Corollary. If the favorable form of an event is certain to happen, then q = 0 and p = 1. EXAMPLE 1. Find the probability of not throwing a sum of five with two dice. If the dice fall 1, 4, or 4, 1, or 2, 3, or 3, 2, the sums will be five. Hence, F = 4, T 6.636, p = 36 , 9 = 8. = = Hence, the probability of not throwing a sum of five with two dice is 8. If a sum of money s is paid upon the happening of an event whose probability is p, the product sp is called the mathematical expectation. EXAMPLE 2. According to the experience of the American life insurance companies, of 100,000 children 10 years of age, 749 die within a year. Neglecting interest on money and the cost of administration, what would be the cost of insuring the life of a 10-year old child for $1000 for one year? = 1000 7.49, and hence the re 749 100,000 The mathematical expectation is quired premium is $7.49. That is, if each of the 100,000 children paid $7.49, then the sum $749,000 resulting would be sufficient to make 749 payments of $1000 each. EXAMPLE 3. From a bag containing 8 white balls and 6 black balls, 5 balls are drawn at random. What is the probability that 3 are white and 2 are black? From 14 balls, 5 can be selected in 14C5 ways. From 8 white balls, 3 can be 2 can be selected in C2 ways. and 2 black balls is selected in 8C3 ways, and from 6 black balls, Hence the probability of drawing 3 white EXERCISES 1. From a bag containing 3 white balls and 7 black balls one ball is taken at random. What is the probability that it will be white? Black? What is the sum of these probabilities? 2. From a bag containing 4 white balls and 8 black balls 2 balls are drawn at random. What is the probability that they will both be black? both white? one white and one black? What is the sum of these probabilities? 3. If two dice are thrown what is the chance of throwing a sum of seven? double sixes? 4. If the probability of an event is, what is the probability that the event will not happen? What is the probWill not be together? 5. Six persons are about to seat themselves in a row. ability that two specified persons will be together? 6. If a prize of $10 is given for drawing a red ball from a bag containing 2 red balls and 6 white, what is the value of the expectation? 7. If of 92,637 people living at the age of 20 there are 85,441 living at the age of 30, what should be the premium for insuring the life of a person of age 20, for $1000 for 10 years, neglecting interest and administrative charges. 136. Compound Events. Sometimes it is convenient to consider an event as made up of two or more simpler events. Thus if two balls are drawn from a bag containing 4 white balls and 5 black balls, the double drawing may be regarded as a compound event made up of two single drawings performed in succession. The component events into which a compound event is resolved are said to be dependent or independent according as the occurrence of one does or does not affect the occurrence of the others. If two balls are drawn from the bag mentioned above, the probability of drawing a white ball the first time is . If a white ball is drawn the first time and not replaced, the probability of drawing a white ball the second time is 3. But if a black ball is drawn the first time and not replaced, the probability of drawing a white ball the second time is. Hence if the ball drawn the first time is not replaced the events are dependent, as the form of occurrence of one does affect the occurrence of the other. If the first ball is drawn and then replaced the probability |