3
Question
If N denotes the number of ways of selecting r objects our of n distinct objects (r≥n) with unlimited repetition but with each object included at least once in selection, then N is equal to
If N denotes the number of ways of selecting r objects our of n distinct objects (r≥n) with unlimited repetition but with each object included at least once in selection, then N is equal to
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Solution
The correct options are
A r−1Cr−n
C r−1Cn−1
Concept: Total number of non-negative integral solution of x1+x2+......+xr=nisn+r−1Cr−1Also, n identical things can be distributed in r groups in n+r−1Cr−1 ways
Let xi(1≤i≤n) be the number of objects selected of the ith type. Since each object is to be selected at least once, we must have xi≥1 and x1+x2+...+xn=r. We have to find number of positive integral solutions of the above equation. Total number of such solutions is r−1Cn−1=r−1Cr−n.
A r−1Cr−n
C r−1Cn−1
Concept: Total number of non-negative integral solution of x1+x2+......+xr=nisn+r−1Cr−1
Also, n identical things can be distributed in r groups in n+r−1Cr−1 ways
Since each object is to be selected at least once, we must have xi≥1 and x1+x2+...+xn=r.
We have to find number of positive integral solutions of the above equation.
Total number of such solutions is r−1Cn−1=r−1Cr−n.
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