Question

8.14. Give a combinatorial proof of the following binomial identity: 3 0) (+) = ---() = 2n-m k= m (Hint: Think of the number of ways of picking two disjoint subsets from a set of cardinality n so that one subset is of cardinality m and the other subset is arbitrary.)

Answer 1

solution : Given that то Рәove : 3 () - -т) Kam poroof consider a set of cardinality in we want to choose two subsets forom this set one with m elements and other subset having any number of from oremaining elements nom elements TO choose m elements foom n elements, we have come I ways 2006 number of SUb8L48 that can be formed from remaining nom elements are anom

(since number of subsets of a set having or elements is 293 Theore foore, number of ways to do task o g is c ) 2n-m Now we can do the same task on a diffeurent way we want two subsets, one having a elements other, any number of elements from nomi elements so this second set can contain 2 elements where L ganges forom o to nam so what we do is we fiorst choose L+m elements from n elements where oslan-m and this can be done in

and then choose m elements from these 2tm elements in Iltm ways I lemn so number of ways = 1 n Lam ( m . where 01 Len-m so Total number of ways to choose two nem m subete - Cu tine Subsets m ltr - 10 2 how put itm=k hanto=7F 0= 7 os 7 -u- u S u=htlu-u) = These foore, The best me) (eon D= CK Lo CZ+m > 3

from equations ② and ③ we get . С. 27 ч. ) - ( x ) ТИ І К - М С 13 — Х Thank you