Permutations and Combinations

Permutations and Combinations

Question

Find sum of integers from 1 to 100 that are divisible by 2 or 5.

Answer

Let space space straight S subscript 1 =  Sum of all the numbers from 1 to 100 which are divisible by 2.
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#6 {main}</pre> =  Sum of all the numbers from 1 to 100 which are divisible by 5.
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#6 {main}</pre> =  Sum of all numbers from 1 to 100 which are divisible by both 2 and 5.

∴           Required Sum = straight S subscript 1 space plus space straight S subscript 2 space minus space straight S subscript 3                                                     ...(i)
Now,                     straight S subscript 1 space equals space 2 plus 4 plus 6 plus........... plus 100
Here,                  space space space space straight a equals 2 comma space space space straight d equals 2 comma space space straight t subscript straight n space equals space 100      
rightwards double arrow          a + (n - 1)d = 1800 rightwards double arrow   2 + (n - 1)2 = 100
or                (n - 1)2 = 98        or       n - 1 = 98 over 2 space equals space 49   or  n = 50

∴                         space space straight S subscript 1 space equals space 50 over 2 left square bracket 2 space plus space 100 right square bracket space equals space 25 space cross times space 102 space equals space 2550
                           straight S subscript 2 space equals space 5 space plus space 10 space plus space 15 space plus space 20 space plus........... space plus space 100
Here, a = 5, d = 5, straight t subscript straight n space equals space 100
rightwards double arrow          a + (n - 1)d = 100       or      5 + (n - 1)5 = 100
rightwards double arrow            n - 1 = 19 rightwards double arrow    n = 20

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#6 {main}</pre>
                 straight S subscript 3 space equals space 10 plus 20 plus 30 plus 40 plus.......... plus 100 space equals space 10 over 2 left square bracket 10 plus 100 right square bracket equals 5 left square bracket 110 right square bracket equals 550
∴  from (i), we have
           Required sum = 2550 + 1050 - 550 = 3050.



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