If some or all of n objects are taken at a time, then the number of combinations is `2^(n) - 1`.

Show that in the number of combinations of 2n different things taken n at a time , the number of combinations in which a particular thing occurs , is equal to the number in which it does not occur .

If the number of combinations of n different things taken (r+r) at a time be equal to the number of combinations taken (r-r) at a time , find the value of n.

STATEMENT-1 :The total number of combinations of n thing by taking some or all is 2^(n) . STATEMENT -2 : 5 identical balls can be distributed among 10 identical boxes in only one way if not more than one ball can go into a box. STATEMENT-3 : In the permutations of n things taken r at a time , the number of permutations in which m particular things occur together is ""^(n - m)P_(r - m) * ""^(r)P_(m)

STATEMENT-1 :The total number of combinations of n thing by taking some or all is 2^(n) . STATEMENT -2 : 5 identical balls can be distributed among 10 identical boxes in only one way if not more than one ball can go into a box. STATEMENT-3 : In the permutations of n things taken r at a time , the number of permutations in which m particular things occur together is ""^(n - m)P_(r - m) * ""^(r)P_(m)

If six times the number of permutations of n things taken 3 at a time is equal to seven times the number of permutations of (n-1) things chosen 3 at a time, find n.

If ""^(n)C _r denotes the numbers of combinations of n things taken r at a time, then the expression ""^(n) C_(r+1) +""^(n)C_(r-1) +2 xx ""^(n)C_r equals

If ""(n)C_(r) denotes the number of combinations of n different things taken r at a time, then the vlaue of ""^(n)C_(r+1)+""^(n)C_(r-1) + 2. ""^(n) C_(r) is-