Suppose there are 4 integers: 1,2,3 and 4

S is even, if m is even

Here m is even (4)

S = 1+1*2+1*2*3+1*2*3*4 = 33

So, S is not even

Hence, option 1 is eliminated

Suppose there are 5 integers: 1, 2, 3, 4 and 5

So, m = 5 = odd

Odd i = 3 ( one, three and five) = odd

S = 1+1*2+1*2*3+1*2*3*4+1*2*3*4*5 = 153 = odd

So, option 2 is eliminated

Suppose there are 5 integers: 1, 2, 3, 4 and 5

n1=1, n2=2, n3=3, n4=4 and n5=5

Odd ni = n1, n3 and n5

Largest value of i for which ni is odd = n5 = 5, which is odd

S = 1+1*2+1*2*3+1*2*3*4+1*2*3*4*5 = 153 = odd

So, option 4 is eliminated

Suppose there are 5 integers: 1, 2, 3, 4 and 5

n1=1, n2=2, n3=3, n4=4 and n5=5

Even ni = n2 and n4

Least value of i for which ni is even = 2, which is even

S = 1+1*2+1*2*3+1*2*3*4+1*2*3*4*5 = 153 = odd

So, the answer is option 3

Suppose there are 4 integers: 1,2,3 and 4

S is even, if m is even

Here m is even (4)

S = 1+1*2+1*2*3+1*2*3*4 = 33

So, S is not even

Hence, option 1 is eliminated

Suppose there are 5 integers: 1, 2, 3, 4 and 5

So, m = 5 = odd

Odd i = 3 ( one, three and five) = odd

S = 1+1*2+1*2*3+1*2*3*4+1*2*3*4*5 = 153 = odd

So, option 2 is eliminated

Suppose there are 5 integers: 1, 2, 3, 4 and 5

n1=1, n2=2, n3=3, n4=4 and n5=5

Odd ni = n1, n3 and n5

Largest value of i for which ni is odd = n5 = 5, which is odd

S = 1+1*2+1*2*3+1*2*3*4+1*2*3*4*5 = 153 = odd

So, option 4 is eliminated

Suppose there are 5 integers: 1, 2, 3, 4 and 5

n1=1, n2=2, n3=3, n4=4 and n5=5

Even ni = n2 and n4

Least value of i for which ni is even = 2, which is even

S = 1+1*2+1*2*3+1*2*3*4+1*2*3*4*5 = 153 = odd

So, the answer is option 3

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