The number of straight lines that can be formed by joining $8$ points on a circle is:

If $P(56, r+6) : P(54, r+3) = 30800 : 1$,find $r$.

$\sum\limits_{r = 0}^m {^{n + r}{C_n} = } $

How many numbers can be formed with the digits $1, 2, 3, 4, 3, 2, 1$ so that odd digits always occupy the odd places?

In an examination,Yamini has to select $4$ questions from each part. There are $6, 7,$ and $8$ questions in Part-$I,$ Part-$II,$ and Part-$III,$ respectively. What is the number of possible combinations in which she can choose the questions?

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