Much distribute, isn't it?

Probability Level pending

The Car's Industry will be distribute r r -much their's products with type : A , B , C , D , E A, B, C, D, E to a certain city. If this city demand to the Industry sent the product A A & D D at least once, then the number of choice to distributes r r -products to this city can be expressed with k r − l × m r + n r k^r-l×m^r+n^r , with k , l , m , n k,l,m,n positive integers. Find k + l + m + n k+l+m+n


The answer is 14.

Muhammad Alif by Muhammad Alif , 20, Indonesia

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1 solution

Muhammad Alif
Jul 16, 2018

Let a k a_k denote the number of r r -object into 5 5 distinct box, with 2 2 -box has at least one object. We can consider the Exponential Generating Function for the a r a_r , is :

( x + x 2 2 ! + x 3 3 ! + . . . ) 2 ( 1 + x + x 2 2 ! + x 3 3 ! + . . . ) 2 (x+\frac{x^2}{2!}+\frac{x^3}{3!}+...)^2(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...)^2

= ( e x − 1 ) 2 ( e 3 x ) =(e^x-1)^2(e^{3x})

= e 5 x − 2 e 4 x + e 3 x =e^{5x}-2e^{4x}+e^{3x} = ∑ r = 1 ∞ ( 5 r − 2 × 4 r + 3 r ) x r r ! =\sum_{r=1}^{∞} (5^r-2×4^r+3^r)\frac{x^r}{r!}

Then a r = 5 r − 2 × 4 r + 3 r a_r=5^r-2×4^r+3^r , and the desired value is 5 + 2 + 4 + 3 = 14 5+2+4+3=14

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