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Permutations & Combinations: Number Ways Biscuits Can Distributed Five Beggars Seated

JEE Maths question with a full step-by-step solution.

Question
If NN is the number of ways in which 1010 biscuits can be distributed to five beggars seated in a row such that no two consecutive beggars remain without biscuit, then N100\dfrac N{100} is
Solution
Answer: 7.71 (± 0.01)
Step 1: The biscuits are identical; the beggars are distinguishable and sit in a fixed row B1,B2,B3,B4,B5B_1,B_2,B_3,B_4,B_5. We must count non-negative integer solutions of
x1+x2+x3+x4+x5=10x_1+x_2+x_3+x_4+x_5 = 10
 in which no two adjacent xix_i are both 00. Step 2: If exactly kk of the five get nothing, those kk seats must be pairwise non-adjacent among the five, and the other 5−k5-k beggars share the 1010 biscuits with each getting at least one. Every distribution has exactly one value of kk, so the cases below are disjoint and cover all of them. Step 3: To choose kk pairwise non-adjacent seats from a row of nn, put the n−kn-k occupied seats down first and place the kk empty ones in distinct gaps among the n−k+1n-k+1 gaps they create, which gives (n−k+1k)\binom{n-k+1}{k}. With n=5n = 5:
k=0: (60)=1,k=1: (51)=5,k=2: (42)=6,k=3: (33)=1k = 0:\ \binom60 = 1 ,\qquad k = 1:\ \binom51 = 5 ,\qquad k = 2:\ \binom42 = 6 ,\qquad k = 3:\ \binom33 = 1
 and k≥4k \ge 4 is impossible (four non-adjacent seats need at least seven). Step 4: Each of the 5−k5-k non-empty beggars gets at least one of the 1010 biscuits, which by stars and bars is (95−k−1)=(94−k)\binom{9}{5-k-1} = \binom9{4-k}:
k=0: (94)=126,k=1: (93)=84,k=2: (92)=36,k=3: (91)=9k = 0:\ \binom94 = 126 ,\quad k = 1:\ \binom93 = 84 ,\quad k = 2:\ \binom92 = 36 ,\quad k = 3:\ \binom91 = 9
 Step 5:
N=1⋅126+5⋅84+6⋅36+1⋅9=126+420+216+9=771N = 1\cdot126+5\cdot84+6\cdot36+1\cdot9 = 126+420+216+9 = 771
N100=771100=7.71\frac N{100} = \frac{771}{100} = 7.71
 (Running over all (144)=1001\binom{14}4 = 1001 non-negative solutions of x1+⋯+x5=10x_1+\cdots+x_5 = 10 and rejecting those with two adjacent zeros confirms 771771.) Answer: 7.717.71
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