A student has three mangos two papayas and two kiwifruits .if the student eats one piece of fruit each day and only the type of fruits matters ,in how many different ways can these fruits be consumed ?

Here, we assume that all the fruits are consumed, that is the process of consumption ends after 7 days. This is a permutation problem since the order in which the fruits are consumed matter.

The total number of fruits $n=3+2+2=7$. There are $n_1=3$ mangoes, $n_2=2$ papayas and, $n_3=2$ kiwifruits.

The number of ways in which these fruits can be consumed is ${n!\over n_1!\times n_2!\times n_3!}={7!\over3!\times2!\times2!}=7\times6\times5=210$ ways.

Therefore, the number of ways in which these fruits can be consumed is 210 ways.