theorem:- The no. of all combinations of n distinct objects taken r at a time is given by `nC_r`= `(n!)/((n-r)!.r!)`

Notation + theorem :- Let r and n be the positive integers such that 1<=r<=n. Then no.of all permutations of n distinct things taken r at a time is given by (n)(n-1)(n-2)....(n-(r-1))

Prove that number of permutations of n distinct objects taken r at a time; when a particular onject is never taken in each a arrrangement is (n-1)P_(r)

Prove that the no.of all permutations of n different objects taken r at a time when a particular object is to be always included in each arrangement is r.(n-1)P_(r-1)

Prove that the number of combinations of n things taken r at a time in which p particular things always occur is .^(n-p)C_(r-p)

Prove that the no.of permutations of n different objects taken r at a time in which two specified objects always occur together 2!(r-1)(n-2)P_(r-2)

Distribution of n distinct object in r different boxes (empty box allowed)

Distribution of n distinct object in r different boxes (empty box not allowed)

Assertion (A) : The number of selections of 20 distinct things taken 8 at a time is same as that taken 12 at a time Reason (R): C(n, r) = C(n, s), if n = r + s

If N denotes the number of ways of selecting r objects of out of n distinct objects (rgeqn) with unlimited repetition but with each object included at least once in selection, then N is equal is a. .^(r-1)C_(r-n) b. .^(r-1)C_n c. .^(r-1)C_(n-1) d. none of these