Multiplying numerator and denominator by (n − r)!, we get it follows that the number of combinations of n things taken r at a time is the same as the number taken n -r at a time. This is evident in any practical example, since the number of ways in which r things may be selected is evidently the same as the number of ways in which the remaining nr things may be discarded. EXERCISES 1. How many dinner parties of 6 can be arranged from among 12 possible persons without having the same company of 6 twice? Ans. 924. 2. How many different assemblages of 1000 persons can be selected from an assemblage of 1003 persons? 3. There are seven letter boxes in a town. How many ways can a person post two letters? Ans. 28. 4. How many straight lines can be drawn through pairs of points selected from 12 points no three of which are in the same straight line? 5. There are 5 trails to the top of a mountain. may a person go up and return by a different trail? Ans. 66. In how many ways 9. How many parallelograms are formed when a set of 9 parallel lines is met by another set of 8 parallel lines? 10. At a meeting of 10 men each shakes hands with all the others. How many handshakes are there? 11. In how many ways can 9 beads of different colors be strung so as to form a bracelet? Ans. 40,320. 12. A Yale lock contains 5 tumblers (cut pins), each capable of being placed in 10 distinct positions. At a certain arrangement of the tumblers, the lock is open. How many locks of this kind can be made so that no two shall have the same key? 13. Two different varieties of corn are planted side by side. How many different kinds will be harvested. Ans. 3. 14. Three different varieties of oats are planted side by side. How many different kinds will be harvested? Ans. 6. 15. A farmer buys two different kinds of thoroughbred chickens and allows them to mix freely. How many different kinds of chickens will he have at the end of the third year of hatching if all stock is sold when one year old? Ans. 21. 16. Out of the 26 letters of the alphabet, in how many ways can a word be made consisting of 5 different letters, two of which must be a and e? Ans. 242,880. 17. How many different throws can be made with 5 dice? 18. From a complete suit of 13 cards, 5 are to be selected which shall include the king or queen, or both. In how many ways can this be done? Ans. 825. 19. Five baseball nines wish to arrange a schedule of games in which each nine shall meet every other nine three times. How many games must be scheduled? Ans. 30. 20. How many odd numbers without repeated digits are there between 3000 and 8000? How many of these are divisible by 5? 21. In how many ways can a pack of 52 playing cards be divided into 4 hands, the order of the hands, but not the cards in the hand, to be regarded? CHAPTER XVI THE BINOMIAL EXPANSION-LAWS OF HEREDITY 207. Product of n Binomial Factors. If the indicated mul tiplications are performed and terms containing like powers of x The number of these terms is the number of combinations that can be made from n a's, 2 at a time, i. e., nC2. The number of these terms is the number of combinations that can be made from n a's, 3 at a time, i. e., nCз. If now each of the a's be replaced by y, it is evident that, (2) ... (x + y)" = x2 + nx"-1y + nС2x-2y2 + nС3x"−3y3 + ··· + nxy”-1 + y”. This is known as the binomial expansion, or binomial formula. 208. Binomial Theorem. If x and y are any real (or imaginary) numbers and if n is a positive integer, then the binomial formula (2) is valid. The following observations will be of value. (1) The exponent of x in the first term is 1 and decreases by 1 in each succeeding term. (2) The exponent of y in the second term is 1 and increases by 1 in each succeeding term. (3) The coefficient of the first term is 1, that of the second term is n. The coefficient of any term can be found from the next preceding term by multiplying the coefficient by the exponent of x and dividing by one more than the exponent of y. (4) The (r+ 1)th term is nCrx”—ˆy”, i. e., The coefficient of this (r + 1)th term is the product of the first r factors of factorial n, divided by factorial r. (5) The sum of the exponents of x and y in any term is n. (6) The number of terms is n + 1. To prove the rule in statement (3) apply it to the (r + 1)th term, but this is precisely nCr+1, which was to be proved. 209. Binomial Coefficients. The coefficients in the binomial expansion are called binomial coefficients. Their values are given in the following table for a few values of n. This table is called Pascal's triangle. TABLE OF BINOMIAL COEFFICIENTS, Cr.-PASCAL'S TRIANGLE n = 10 n = 11 etc. 10 45 120 210 252 210 120 45 105 11 55 165 330 462 462 330 165 55 11 etc. etc. NOTE. If any number in the table be added to the one on its right, the sum is the number under the latter. 210. Sum of Binomial Coefficients. A great many uses for binomial coefficients and a great many relations among them have been discovered. Two of these are as follows. (1) The sum of the binomial coefficients of order n is 2". We verify from the above table that |