Permutation and combination

Answer :

^n–1C₅ + ^n–1C₆ < ⁿC₇

                                      ∴       ⁿC₆ < ⁿC₇

                                      ∴       [n! / {6! (n – 6)!}] < [n! / {7! (n – 7)!}]

                                      ∴       [1 / (n – 6)] < (1/7)

                                      ∴       n – 6 > 7

                                      ∴       n > 13

                                       ∴       least value of n = 14

Answer :

Total arrangement = n!

                   Two particular books are together = (n – 1)!2

                   Two particular books are not together = n! –[(n – 1)! 2]

                   ∴       (n – 2) (n – 1)! = 480

                   ∴       (n – 2) (n – 1)! = 4 × 120

                   ∴       (n – 2) (n – 1)! = (6 – 2) (6 – 1)!

                   ∴       n = 6

                   ∴       No of books on shelf = 6

Answer :

digits 1, 3, 5, 7, 9. Hence there will be 120 number to add.

                   Each vertical column of digits contains equal number of 1’s 3’s 5’s and 7’s

                   9’s .

                   ∴       The sum of each column = (120 ÷ 5) × (1 + 3 + 5 + 7 + 9) = 600

                   First column represents 10000’s. 2^nd column 1000’s etc.

                   ∴       sum is (600 × 10000) + (600 × 1000) + (600 × 100) +(600 × 10)

                                                                                       + (600 × 1)

                   = 600 [10000 + 1000 + 100 + 10 + 1]

                   = 600 × (11111) = 6666600

Answer :

In MATHEMATICS, M – 2 times, T – 2 times and A – 2 times, H, E, I, C, S,

                   The words with two letters alike = ³C₂ × [(4!) / (2! 2!)] = 18

                   The words with two letters alike and two distinct = ³C₂ × ⁷C₂ × (4! / 2!)

                                                                                                          = 756

                   The words with all letters distinct = ⁸C₄ × 4! = 1680

                   ∴       Total words          = 18 + 756 + 1680

                                                = 2454

Answer :

ⁿ∑_r=0 [r / (ⁿC_r)] = [0 / (ⁿC₀)] + [1 / (ⁿC₁)] + [2 / (ⁿC₂)] + ------ [n / (ⁿC_n)]         (1)

                   also

                   ⁿ∑_r=0 [r / (ⁿC_r)] = [n / (ⁿC₀)] + [(n – 1) / (ⁿC₁)] + [(n – 2) / (ⁿC₂)]

                                                                                      + ------ [0 / (ⁿC_n)]   (2)

                   (1) + (2) gives

                   2 ⁿ∑_r=0 [r / (ⁿC_r)] = [n / (ⁿC₀)] + [n / (ⁿC₁)] + [n / (ⁿC₂)] +------ [n / (ⁿC_n)]

                                          = n {[1 / (ⁿC₀)] + [1 / (ⁿC₁)] + [1 / (ⁿC₂)] + ------ [1 / (ⁿC_n)]}

                   ∴       2 ⁿ∑_r=0 [r / (ⁿC_r)] = n ⁿ∑_r=0 [1 / (ⁿC_r)]

                                                    = n a_n ------------ a_n = ⁿ∑_r=0 [1 / (ⁿC_r)]

                   ∴       ⁿ∑_r=0 [r / (ⁿC_r)] = [(n ∙ a_n) / 2]