Sita has 4 different and simmi has 7 different toys Number of ways in which they can exchange their - Maths - Probability

Question

Sita has 4 different and simmi has 7 different toys Number of
ways in which they can exchange their toys so
that each keeps her inital number of toys, is

Vimala · Accepted Answer

Sita has 4 different toys and Simmi has 7 different toys.

The maximum number of toys they can exchange is 4 and the minimum is 1.

For the exchange of one toy:

Sita can choose in ${}^{4}{C_{1}}$ ways.

Simmi can choose in ${}^{7}{C_{1}}$ ways.

Since both of them have to exchange their toys, the total number of ways for exchanging one toy is ${}^{4}{C_{1}} \times {}^{7}{C_{1}}$ ways.

For the exchange of two toys:

Sita can choose in ${}^{4}{C_{2}}$ ways.

Simmi can choose in ${}^{7}{C_{2}}$ ways.

Since both of them have to exchange their toys, the total number of ways for exchanging two toys is ${}^{4}{C_{2}} \times {}^{7}{C_{2}}$ ways.

For the exchange of three toys:

Sita can choose in ${}^{4}{C_{3}}$ ways.

Simmi can choose in ${}^{7}{C_{3}}$ ways.

Since both of them have to exchange their toys, the total number of ways for exchanging three toys is ${}^{4}{C_{3}} \times {}^{7}{C_{3}}$ ways.

For the exchange of four toys:

Sita can choose in ${}^{4}{C_{4}}$ ways.

Simmi can choose in ${}^{7}{C_{4}}$ ways.

Since both of them have to exchange their toys, the total number of ways for exchanging four toys is ${}^{4}{C_{4}} \times {}^{7}{C_{4}}$ ways.

Therefore the total number of ways for exchanging their toys is ${}^{4}{C_{1}} \times {}^{7}{C_{1}}$ + ${}^{4}{C_{2}} \times {}^{7}{C_{2}}$ + ${}^{4}{C_{3}} \times {}^{7}{C_{3}}$ + ${}^{4}{C_{4}} \times {}^{7}{C_{4}}$ .

Therefore the total number of ways for exchanging their toys is 28 + 126 + 140 + 35.

Therefore the total number of ways for exchanging their toys is 329.

Sita has 4 different and simmi has 7 different toys. Number of ways in which they can exchange their toys so that each keeps her inital number of toys, is

Sita has 4 different and simmi has 7 different toys. Number of ways in which they can exchange their toys so

that each keeps her inital number of toys, is