Question

Let n(A) = 4 and n(B)=k. The number of all possible injections from A to B is 120. then k=​

Answer 1

k = 5

• It is given that n(A) = 4 and n(B) = k.
• Now an injection is a bijection onto its image. Thus we can find the number of injections by counting the possible images and multiplying by the number of bijections to said image.
• Now the number of bijections is given by p!, in which p denotes the common cardinality of the given sets.
• So the required number is $(\left { {{q} \atop {p}} \right. )*p!$ where n(A) = p and n(B) = q.
• Here p = 4 and q = k.
• Now $(\left { {{k} \atop {4}} \right. )*4! = 120$
• this implies $(\left { {{k} \atop {4}} \right. ) = 5$ or k = 5
• Hence the value of k is 5.