Set, Relation and FunctionHard
Question
Let X = {1, 2, 3, 4, 5}. The number of differentordered pairs (Y, Z) that can be formed suchthat 
and 
is empty, is :
and is empty, is :Options
A.53
B.52
C.35
D.25
Solution
y and z
case 1 set y contain no elemets and set Z contain either φ or 1,2,3,4,5 elements.
1 × {5C0 + 5C1 + ......+ 5C5} = 32
case 2 Set y contain 1 elements then
5C1 × (4C0 + 4C1 + 4C2 + 4C3 +4C4) = 80
case 3 set y contain 2 elements, then
5C2 × (3C0 + 3C1 + 3C2 + 3C3 + 3C4) = 80
case 4 set y contains 3 elements the
5C3 × (2C0 + 2C1 + 2C2) = 40
case 5 set y contains 4 elements then
5C4 × (1C0 + 1C1) = 10
case 6 set y contains 5 elements, then 5C5 × 1 = 1
32 + 80 + 80 + 40 + 10 + 1 = 243 = 35
case 1 set y contain no elemets and set Z contain either φ or 1,2,3,4,5 elements.
1 × {5C0 + 5C1 + ......+ 5C5} = 32
case 2 Set y contain 1 elements then
5C1 × (4C0 + 4C1 + 4C2 + 4C3 +4C4) = 80
case 3 set y contain 2 elements, then
5C2 × (3C0 + 3C1 + 3C2 + 3C3 + 3C4) = 80
case 4 set y contains 3 elements the
5C3 × (2C0 + 2C1 + 2C2) = 40
case 5 set y contains 4 elements then
5C4 × (1C0 + 1C1) = 10
case 6 set y contains 5 elements, then 5C5 × 1 = 1
32 + 80 + 80 + 40 + 10 + 1 = 243 = 35
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