Explanation

(a)
If the function is one-one, then select any three from the set $B\ in\ \ ^{7}C_{3}$ ways $i.e.$, 35 ways.
If the function is many-one, then there are two possibilities. All three corresponds to same element number of such functions $= \ ^{7}C_{1} = 7$ ways. Two corresponds to same element. Select any two from the set $B$. The lerger one corresponds to the larger and the smaller one corresponds to the smaller the third may corresponds to any two. Number of such functions $= \ ^{7}C_{2} \times 2 = 42$
So, the required number of mappings $= 35 + 7 + 42 = 84$