Ex. 52. coins: a farthing, a penny, a sixpence, a shilling, a half-crown, a crown, a half-sovereign, and a sovereign? 22. Out of 17 consonants and 5 vowels, how many words can be formed, each consisting of 2 vowels and 3 consonants? 23. How many words of 6 letters may be formed out of 24 letters of the alphabet, with 2 of the 5 vowels in each word? 24. The No. of Perms. of n things, 3 together, is 6 times the No. of Combs. 4 together; find n. 25. The No. of Perms. of n things taken r together is equal to 10 times the No. when taken r—I together; and the No. of Combs. of n things taken r together is to the No. when taken r- I together as 53; required the values of n and r. 26. If, generally, Cm denote the No. of Combs. of m things taken p together; show that C,C+C(-1). 27. A person wishes to make up as many different parties as he can out of 20 friends, each party consisting of the same number; how many should he invite at a time? 28. When the No. of Combs. of 2n things taken r together is the greatest possible; required r. 29. There are 4 regular polyhedrons marked, each face with a different symbol, and the numbers of their faces are 4, 6, 8, 12 respectively; taking all of them together, how many different throws are possible? 30. Find the No. of different Combs. of n things, of which pare of one sort, q of another, r of a third, and so on, when taken 1, 2, 3, &c. n together severally. BINOMIAL AND MULTINOMIAL THEOREMS. 1. In the expansion of (a+x)", the (r+1)th term is n(n−1)(n−2). . . (n−r+1) ̧n-rxr. 1.2.3 .r 2. In the expansion of (a+bx+cx2+...+kxt)", the term involving xm is 1.2.3...n (1.2.3..p)(1.2.3..q)(1.2.3..r)&c. where p+q+r+...=n, and q+2r+...=m. Ex. 53. -ap.bq.c2.&c.x+2r+.. 1. Expand (1+x)7 ; (1+2x)5; (1+2) 0 ; 6 Ex. 53. -5 2. Expand (1-34); (1-)'; (---) ̄*; (++)"'. 8. Expand (h+k√—1)7; (b−y√—1)3 ; (−a3 + ∞ √ − 1)". 9. Expand (1+x+x2)5; (1-2x+x2)3; (a-2b+3c)4. Find in the following binomials or multinomials expanded— 24. The 6th term of (4a2cx—3c3y3); and of (ax—bx2 √√ − 1)}. 25. The 5th term of (3x-2y)-10; and of (−s+t √ −1)3. 26. The (r+1)th term of {xy—(3yz)1⁄2}4. 27. The greatest term of (1 +§)3. Ex. 53. Find in the following binomials or multinomials expanded,23. The middle term of (1+r)"; and of (1+x+x2)12. 29. The middle term of (2-5x-7x2+x3 + 3x4)5. 30. The No. of terms in (a+b+c); and in (a + bx + cx2+ dx3)4. 31. Show that (x + 1)TM" = (x2 + 131n) + 2n (22n—2. x2n-2 I 2n(2n-1) + + x2n-4. 1.2 +... 35. a−(a+b)n+(a+2b)n (n − 1) — (a+3b)n(n−1) (n−2) +... 1.2 1.2.3 36. If the coefficients of the (r+1)th and (r+3)th terms of (1+x)" are equal, n being a positive integer; find r. 37. The coefficients of x in the 5th and 7th terms of (1+2x)" are 1120 and 1792 respectively; find n. 38. The coefficients of x in the 3rd and 5th terms of (1-x)" are and respectively; find n. Ex. 53. 39. If generally n, be the coefficient of the (r+ 1)th term of (1+x)”, show that (n+p)r=nr+Nr−1Px+Nr−2P2+ &c. +n1Pr−1+Pr• 40. If generally m, be the coefficient of the (r+1)th term of (1-x)-m, show that m,+ (m+1),−1=(m+I),. 41. Find the sum of the squares of the coefficients in the expansion of (1+x)", when n is a positive integer. N.B. Equate the coeffts. of an in (1+x)". (x+1)" and in (1+2)2n. 42. Find the sum of the products of every two consecutive coefficients in the expansion of (1+x)", n being a positive integer. N.B. Equate the coeffts. of 2-1 or an+1 in (1+x)". (x+1)" and in 43. If a, b, c, d be any consecutive coefficients of an expanded binomial, show that (bc+ad) (b−c)=2(ac2—b2d). 44. If s= sum of two quantities, p=their product, and q=the quotient; show that p2=s^(q3—493 +' 4.5.6 4-594. q5+ &c. 1.2 1.2.395 ·.). 45. If s= sum of the squares of any two quantities, p=2x product, and P= the pth power of the sum; show that Ex. 54. Expand in a series of ascending powers of x Ex. 55. Find the value of x in an infinite series, in terms of y— Ex. 56. Find, by the method of Indeterminate Coefficients, the sum→ 1. Of 1+2+3+4+ &c. to II terms. 2. Of 12+42+72+102+&c. to n terms. 4. Of 1.2 +3.4+5.6+&c. to n terms. 7. Of 13+23+33+ &c. to 20 terms. 8. Of 13+33+53+&c. to n terms. 9. Of 15 terms of a series whose nth term is (2n−1)(3n+1). |