Three distinct numbers are selected randomly from the set {1, 2, 3,$\ldots$$\ldots$, 40}. If the probability, that the selected numbers are in an increasing G.P. is m n , gcd(m,$n) = 1$, then$m + n$is equal to _____.
Step-by-Step Solution
Key Concept: Compute the favourable cases for probability from geometric progression and divide by the total equally likely cases.
1$\le$$a < ar < ar$2$\le$40 (4949) (If r$\ in $N ) If$r = 2$1$\le$$a < 2a < 4a$$\le$40 a$\ in ${1,$\ldots$$\ldots$. , 10} ________ (10 GP) If$r = 3$1$\le$$a < 3a < 9a$$\le$40 a$\ in ${1, 2, 3, 4} ________ ________ (4 GP) If$r = 4$1$\le$$a < 4a < 16a$$\le$40 a$\ in ${1, 2} ________ ________ (2 GP) If$r = 5$1$\le$$a < 5a < 25a$$\le$40 a$\ in ${1} ________ ________ (1 GP) If$r = 6$1$\le$$a < 6a < 36a$$\le$40 a$\ in ${1} ________ ________ (1 GP)$(P = 18$$9880 = 4940$9 ) as per NTA for r$\ in $N$m + n = 4949$If –r $\notin$ N (also possible) 3$r = 2$2 9a ar = ;$a = 4k$4 (4, 6, 9) ⎫ ⎪ ⎪ ⎪ ⎪ (8, 12, 18) ⎬ 4GP (12, 18, 27) ⎪ ⎪ ⎪ ⎭ ⎪ (16, 24, 36) 5 2 25a$r = ar$= ;$a = 4k$2 4 (4, 10, 25)$\ldots$$\ldots$. . (1)GP 4 2 16a$r = ar$=$\to$$a = 9k$3 9 (9, 12, 16), (18, 24, 32)$\ldots$$\ldots$. (2)GP 5 2 25a$r = ar$= ;$a = 9k$3 9 (9, 15, 25)$\ldots$$\ldots$$\ldots$. (1)GP 5 2 25a$r = ar$= ;$a = 16k$4 16 (16, 20, 25)$\ldots$$\ldots$$\ldots$. . (1)GP 6 2 36a$r = ar$= ;$a = 25k$5 25 (25, 30, 36)$\ldots$$\ldots$$\ldots$. . (1)GP$Total = 18 + 10 = 28$28 28 7 P =$C_{3}$= = 40 9880 2470$m + n = 2477$
Correct Answer: 4949