In how many ways can $21$ English and  $19$ Hindi books be placed in a row so that no two Hindi books are together

If ${ }^{n} P_{r}={ }^{n} P_{r+1}$ and ${ }^{n} C_{r}={ }^{n} C_{r-1}$, then the value of $r$ is equal to:

Find the number of ways in which two Americans, two British, One Chinese, One Dutch and one Egyptian can sit on a round table so that person of the same nationality are separated?

There were two women participating in a chess tournament. Every participant played two games with the other participants. The number of games that the men played between themselves proved to exceed by $66$ the number of games that the men played with the women. The number of participants is

The number of arrangements of the letters of the word $SATAYPAUL$ such that no two $A$ are together and middle letter is consonant, is