The number of ways of selecting 8 books from a library which has 9 identical books each on mathematics, physics, chemistry and english is? Answer is 165. Pls solve

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Your question is quite good. Now we will denoteMathematicsMPhysicsPChemistryCEnglishENow as each books of each subject is identical, hence number of ways to select 1 book or 2 books or.......or 8 books of a particular subject will be 1.As there is no restriction, hence any combination is possible i.e.All 8 books can be of E, or no books can be of E in the selected books and so on, hence 0E80P80C80M8Now here upper limit is 8 because we have to select 8 books not 9.So our objective is to find number of integral solutions of:E+P+C+M=8  Where 0E8 and 0P8 and 0C8 and 0M8Now number of integrals solutionsN ofx1+x2+x3+....xr=n Where 0x1n and 0x2n and 0x3n and........ 0xrnis given by:N=n+r-1Cr-1Here n=8 and r=4HenceNumber of ways=8+4-1C4-1=11C3= Answer

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