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Question

If 4 integers are to be selected from {1,2,3,......20} such that the sum of the integers should be the multiple of 4, then number of ways to select the numbers are

A
1220
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B
1120
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C
1200
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D
970
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Solution

The correct option is A 1220
Every integer in the set is of the form (4p,4p+1,4p+2,4p+3)
We can either select 4 integers of the same type say 4p then no. of ways are = 204C4
same is possible for every type
no. of ways = 4(204C4)=4 (5C4)=20

anothers ways is to select two integers of 4p type and two integers of (4p+2) type then no. of ways are
= 5C2× 5C2=(5C2)2=100
and to select two integers of (4p+1) type and two integers of (4p+3) type
no. of ways are = 5C2× 5C2=(5C2)2=100

One more way is to select two integers of 4p type and one integer of (4p+1) type and one integer of (4p+3) then no. of ways are
= 5C2× 5C1× 5C1=250
Also to select two integers of (4p+2) type and one integer of (4p+1) type and one integer of (4p+3) no. of ways are
= 5C2× 5C1× 5C1=250
​​​​​Also to select two integers of (4p+3) type and one integer of (4p) type and one integer of (4p+2) no. of ways are
= 5C2× 5C1× 5C1=250
Also to select two integers of (4p+1) type and one integer of (4p+2) type and one integer of (4p) no. of ways are
= 5C2× 5C1× 5C1=250

So total no. of ways are = 20+100+100+250+250+250+250=1220

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