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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?

 Nov 25, 2020
 #1
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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%

 Nov 25, 2020
 #2
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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.

Guest Nov 25, 2020

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