516 views
5 votes
5 votes

1- Determine the number of 5-combinations of { 1.a , ∞.b ,∞.c , 1.d }. 

2-More generally , develop a formula for the number of r-combinations of a collection of letters a1 , a2 , ..., awhose repetition numbers are either 1 or ∞.

P.S. i got the answer of 1st but unable to generalize it as asked in 2nd.

1 Answer

1 votes
1 votes
Let break it into two parts one set X containing element which can be used only ones i.e 1.a, 1.b , another set Y contains all elements which have unlimited repeatation like b,c. r-combination means choose r elements from these two.

We have several choices like pick 1 element from set X and rest r-1 from Y, another case is choose 2 elements from X and rest (r-2) from Y and so on....upto choosing 0 elements from X and all from Y.

Choosing from X set is simple choosing j element from x elements, and r-j remaining from set Y is choosing like " (r+n-1)C(r) where we need r-combination from n element which have infinite copies".

So atlast we sum up all the cases :

$\sum_{j}^{j\leq x}\left ( \binom{x}{j} \binom{(r-j+y-1)}{r-j}\right )$

Please tell me if there is any mistake, I have just tried to do it.

Related questions

2 votes
2 votes
0 answers
1
Ankita Choudhary asked Jul 3, 2017
425 views
Twenty athletes compete in a contest. Each of 3 judges assign 20 different ratings to the 20 athletes. For an athlete to be named winner he must be given the highest rati...
0 votes
0 votes
0 answers
2
kidussss asked Jul 8, 2022
459 views
Solve the recurrence relation $a^{2}n-5a^{2}_{n-1}+4a^{2} _{n-2}=0$, if $a_{0}=4, a_{1}=13, n>1$
0 votes
0 votes
0 answers
3
Swarnava Bose asked Jun 8, 2023
211 views
What is the total number of integer partitions ( unordered Summation) of the natural number 8 ?I am getting 22. Is it correct ?
0 votes
0 votes
1 answer
4
Swarnava Bose asked Jun 5, 2023
235 views
Given there are 3 full baskets of apples, mangoes, and oranges. How many ways possible ifa) You need to buy any 4 fruits out of these 3 baskets ?b) you buy any 4 fruits s...