Two distinct positive integers from 1 to 50 inclusive are chosen. Let the sum of the integers equal S and the product equal P. What is the probability that P*S is one less than a multiple of 5?
There maybe an easier way of doing it, but I don't know of any!.
50 C 2 =1,225 number of ways of picking 2 distinct numbers out of 50. This gives rise to 1225 sums and 1225 products. You multiply their sums by their products and you get 1225 products.
1225 products mod 5 ==4. ANY number that ends in 4 or 9 is one LESS than a multiple of 5. Out of 1225 such numbers, my computer lists the following integers:
(49, 99, 299, 364, 544, 684, 684, 779, 1029, 1144, 1144, 1334, 1449, 1449, 1584, 1584, 1584, 2184, 2184, 2184, 2184, 2349, 2604, 2744, 2744, 3844, 4224, 4224, 4224, 4224, 4719, 4884, 5304, 5304, 5304, 6624, 6624, 7104, 7104, 7104, 7104, 7714, 7904, 7904, 8424, 8424, 8619, 9009, 9009, 10619, 10824, 10824, 10824, 10824, 11424, 11424, 11844, 12384, 12384, 12384, 12384, 12384, 13464, 13464, 16744, 16744, 17484, 17719, 18189, 18424, 19344, 19584, 19584, 20064, 20064, 21609, 24684, 25194, 25194, 25704, 25704, 25704, 26364, 26624, 26884, 27144, 27404, 27664, 27664, 29574, 31104, 31104, 31104, 36064, 36064, 38019, 38304, 38304, 38304, 38304, 38874, 39729, 40774, 41354, 42224, 43424, 43424, 43719, 44604, 44604, 44604, 49654, 49959, 51894, 55944, 56259, 56889, 59584, 59904, 59904, 65274, 65934, 66924, 69144, 71944, 72964, 73304, 73984, 77004, 78039, 84064, 84419, 85484, 85839, 87984, 88704, 89424, 89784, 91834, 94024, 95904, 97384, 98124, 105944, 106704, 111804, 112189, 116064, 121344, 129519, 137104, 156864, 161994, 162864, 164604, 187824, 193844, 194304, 228144) >>Total = 159 such numbers. Therefore, the probability is:159 / 1225 =12.98%
Sorry, I counted some numbers twice. This new list is more accurate:
84 264 324 624 714 1134 1254 1794 1944 2604 2784 3564 3774 4674 4914 5934 6204 7344 7644 384 884 1584 2484 3584 4884 6384 8084 9984 1224 2184 2464 3744 4104 5704 6144 8064 8584 10824 11424 13984 14664 17544 18304 21504 22344 2574 4374 6624 9324 12474 16074 20124 24624 4914 7254 7904 10764 11544 14924 15834 19734 20774 25194 26364 31304 32604 38064 39494 8064 11914 16464 21714 27664 34314 41664 12654 16974 18144 23184 24534 30294 31824 38304 40014 47214 49104 57024 59094 18354 25004 32604 41154 50654 61104 25944 32844 34684 42504 44574 53314 55614 65274 67804 78384 81144 34944 45144 56544 69144 82944 46284 56364 59024 70224 73164 85484 88704 102144 105644 59334 73834 89784 107184 75174 89034 92664 107844 111804 128304 132594 93024 112574 133824 114114 132354 137104 156864 161994 137514 162864 164604 187824 193844 194304 228144 >>Total = 145 such numbers.
The probability =145 / 1225 =29 / 245.
