EXAMPLE 1. A committee of 5 is to be chosen from 7 lawyers and 6 physicians. How many committees will contain (a) just 3 lawyers, (b) at least 3 lawyers? (a) The number of ways of selecting three lawyers is Hence the number of committees which will contain exactly three lawyers is 7C3 × 6C2 35 x 15 525. (b) At least three lawyers are present in each committee of the types: 3 lawyers and 2 physicians, 4 lawyers and 1 physcian, and 5 lawyers. Hence the number of committees which contain at least three lawyers is 7C3 × 6C2 + 7C4 × 6С1 +7 С5 = 756. EXAMPLE 2. How many arrangements can be made consisting of two vowels and three consonants chosen from the letters of the word triangle? The vowels may be selected in 3C2 3 ways. The consonants may be selected in ¿Ñ3 3 10 ways. Hence the total number of selections consisting of two vowels and three consonants is 3С2 × 5C3 = 30. Each of these selections can be arranged in 5! ways. Hence the required number of arrangements is 5! 3C2 × 5C5 EXERCISES 3600. 1. How many alloys can be made from thirty of the known metals chosen two at a time counting one alloy only for each pair of metals? Solve the problem if there are three metals in each alloy. 2. In how many ways can a basket ball team be selected from 9 candidates? If A plays center in every combination, in how many ways can the team be chosen? 3. How many arrangements can be made of 3 vowels and 4 consonants chosen from 5 vowels and 8 consonants? 4. How many straight lines are determined by (a) 5 points no 3 of which are in the same straight line? (b) n points, no 3 of which are in the same straight line? 5. In how many ways can a baseball nine be chosen from 13 candidates, provided A, B, C, D are the only battery candidates, and can play in no other position? 6. In how many ways can a committee of 5 be chosen from 7 democrats and 7 republicans, so that there will be (a) three democrats, (b) no more than three democrats, (c) at least three democrats on the committee? 134. The Binomial Expansion. In finding the product of the binomial factors (x+α1) (x + α2)(x+as), each partial product is obtained by choosing one and only one term from each factor and multiplying the three terms together. The sum of the partial products gives the desired product. There is only one term containing x3, since the three x's can be chosen from the three factors in but one way. The terms of the product containing 22 are obtained by choosing x from two of the factors and an a from the third factor, which gives the partial products x2a1, x2a2, x2as. The number of such terms will be the number of ways we can choose an a from the three a's, or C1. To obtain the term in x, we choose an x from one binomial and two a's from the two remaining in all possible ways, which gives the partial products xa1A2, XA1A3, XA2α3. The number of such products is therefore 3C2. There is only one way of choosing the three a's. (x + α1)(x + α2)(x + α3) If we let α1 x3 + (α1 + A2 + ɑ3)x2 + (α1α2 + α1αз + α2α3)X + α1ɑ2α3. a 2 a3 a, we have 3 (x + α)3 = x3 + 3ax2 + 3a2x + a3. And since, from the preceding, the coefficient of x2 is the number of ways we can choose an a from the three a's, the coefficient of x is the number of ways we can choose two a's out of three a's, and so on, we can write the expansion in the form 3 (x + a)3 = x3 + 3С1ax2 + 3C2a2x + 3C3a3. In a similar manner it can be shown that when n is a positive integer, we have The binomial expansion: n (x + a)" = x2 + nС1axn−1 + n C2α2xn−2 + +nCrarxn~r + +nCna", The values of the coefficients are given in the following table for several values of n. The table is called Pascal's triangle. 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 2′′ – 1. In the binomial expansion for (x + a)" let xa 1. Hence Therefore = 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 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! 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 nCr + nCr-1 = n+1Cr. Compare this with the rule for finding numbers in Pascal's triangle. 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 3ths. 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 ,, & of the total number of trials. 1 2 3 ¿, 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. |