7. How many different hands of 13 cards each may dealt from a deck of 52 cards? be 8. From 8 Republicans and 6 Democrats, how many different committees may be selected, each composed of 3 Republicans and 2 Democrats? 9. From 15 consonants and 5 vowels how many words containing 3 consonants and 2 vowels may be formed, if any arrangement of the letters is considered a word? 10. Fifty points are situated in a plane, no three of the points in a straight line. How many triangles may be formed by connecting the points with straight lines? 11. In a house, 12 windows are available for watching a parade, 6 on the second floor and 6 on the third. If 5 of the windows are occupied, 3 on the second floor and 2 on the third, in how many ways may 7 more persons be accommodated, one at each window? 12. If C2"C5" find the value of n. = 13. How many different sums may be paid with a cent, a 5-cent piece, a dime, a quarter and a half dollar? 14. How many words may be formed from the word Edgar, if each word begins with E and any arrangement of the letters is considered a word? 190. To find the number of permutations of n things, taken n at a time, when they are not all different. Suppose that there are n letters, a, b, c, d, etc., of which a occurs r times, b occurs p times, c occurs q times, and the other letters occur only once. Let N stand for the required number of permutations. It is evident, that in any arrangement of letters, the interchange of like letters will not change the form of the arrangement. Hence, if the r letters a, were all different instead of alike, since r different letters haver permutations, the actual number of permutations of the n letters would be Nxr. Similarly, if the p letters b were all different, since p letters have P permutations, the actual number of permutations of the n letters would be NXrXP. And if the q letters c were all different, the number of permutations of the n letters would be N Xr xp xq. Since, in that case, the n letters are all different, P," = n. = n. That is, The number of permutations of n things, taken all at a time, when r of them are of one kind, p of another, and q of another, and so on, is n p q EXERCISE 140 1. How many permutations may be made with the letters of the word Connecticut, taken all together? n = 11, there are 3 c's, 2 n's, and 2 t's. 11 X 10 X 9 X 8 X 7 X6 X5 X 4 . N = 11 = 2. How many permutations may be made with the letters of the word Roosevelt taken all together? 3. How many different signals can be made with 3 yellow flags, 2 red flags, 5 green flags and 4 blue flags, used all together? 4. How many different numbers can be formed by using the six figures, 1, 1, 2, 2, 3, 3? 191. To find the total number of combinations of n different things. The total number of combinations of n different things is the number of combinations of n different things taken successively, 1, 2, 3, . . . . n at a time. Thus, the total number of combinations of 2 things is, C} + C2 = 2 + 1 = 3, which may be expressed 2a — 1. The total number of combinations of 3 things is C2 + C2 + C2 = 3 + 3 + 1 = 7 or 23 — 1. The total number of combinations of 4 things is, C1 + C + C + C1 = 4 + 6 + 4 + 1 = 15 or 24 — 1, and so on. Hence: The total number of combinations of n different things is equal to 2" - 1. EXERCISE 141 1. How many different sums may be made with a cent, a dime, a quarter, a half dollar and a dollar? 2. How many signals may be made with 7 different flags? 3. How many different quantities may be weighed with the weights, oz., 1 oz., lb. 1 lb. 2 lbs. 5 lbs.? 4. A man belongs to a club of ten members. In how many different ways may he invite one or more of the members to dinner? 192. To find the number of circular permutations of n different things taken all together. If 5 letters a, b, c, d, e were placed in position around a circle in the order abcde, the arrangement could be read abcde, bcdea, cdeab, deabc, eabcd. Hence one circular ar rangement corresponds to five straight line arrangements, that is, the number of circular permutations of 5 different things is one-fifth of the number of permutations of 5 different things taken all together. Similarly, the number of 1 circular permutations of n things will be th the number n of permutations of n different things taken all together. 1. In how many ways may 5 persons seat themselves around a table? 2. In how many orders may 4 couples seat themselves around a table? 3. In how many ways may 4 gentlemen and their wives seat themselves around a table so that each gentleman sits opposite his wife? 4. In how many ways may 4 gentlemen and their wives seat themselves around a table so that each gentleman sits opposite a lady? 5. In how many ways may the colors, violet, indigo, blue, green, yellow, orange and red, be arranged on a disk, the colors radiating from the center? REVIEW EXERCISE XXVII 1. In how many ways may 10 be thrown with two dice? 2. In how many different ways may 15 balls be arranged in a row, if 7 of the balls are wh te, 5 black and 3 red? 3. How many different sums can be paid with a cent, a nickel, a dime, and a quarter? 4. In how many different ways may six students be seated in ten seats? 5. How many baseball nines can be made up of 12 players, the pitcher and catcher of each nine being the same? 6. How many different committees, each consisting of 2 Republicans and 3 Democrats, can be formed from 14 Republicans and 21 Democrats? 7. If 8 points lie in a plane and no three of them in a straight line, how many straight lines can be drawn connecting two points? 8. In a party of 4 ladies and 3 gentlemen, a game of tennis is to be arranged, each side consisting of one gentleman and one lady. In how many ways may this be done? 9. How many different signals can be given with 4 red flags, 3 blue flags, and 3 white flags, each signal requiring all flags? 10. In how many ways can one mail 3 letters in a village containing 5 letter boxes? 11. How many numbers of four figures each can be formed from the figures, 1, 2, 3, 4, 5, 6, 7? 12. Supposing dice were constructed with 10 faces, how many different throws could be made with three dice? 13. A guard of 5 men must be selected every night out of a detachment of 32 men. For how many nights can a different guard be selected and how many times will each man serve? - 14. In how many ways may 4 gentlemen and 4 ladies sit at a round table, so that no two gentlemen sit together? 15. A bus can accomodate 5 passengers on each side. In how many ways can 10 persons take seats, when a certain two always sit on one side and a certain one always on the other? 16. In a certain school, 20 boys come out for a practice game of football. In how many ways may the coach select a trial eleven? After the eleven is selected, he assigns 4 boys to play the four positions back of the line, but the 7 boys in the line he decides to interchange among the seven positions. In how many ways may the team be arranged? 17. A man has 5 coats, 6 vests, and 8 pairs of trousers. In how many different suits can he appear? 18. In how many ways may 7 books be arranged on a shelf, so that two particular books will not be together? 19. How may words can be formed from the letters of the word education, provided the second, fourth, sixth, and last letters are always consonants? 20. In how many different ways can 4 bridge hands of 13 cards each be dealt from a pack of 52 cards? |