In how many ways can we distribute 10 identical looking pencils to 4 students so that each student gets at least one pencil?
To determine the number of ways to distribute 10 identical pencils to 4 students, ensuring each student receives at least one pencil, you can use the Stars and Bars method with a slight adjustment.
Here’s the approach:
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Ensure each student gets at least one pencil: Initially, give one pencil to each of the 4 students. This leaves you with (10 - 4 = 6) pencils to distribute.
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Distribute the remaining pencils: Now, you need to distribute these 6 remaining pencils among the 4 students without any restrictions. This is a classic Stars and Bars problem.
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Apply the Stars and Bars theorem: According to the theorem, the number of ways to distribute (n) identical objects into (k) distinct groups is given by the binomial coefficient (\binom{n + k - 1}{k - 1}) or equivalently (\binom{n + k - 1}{n}).
In this case, (n = 6) (the remaining pencils) and (k = 4) (the number of students). So, the number of ways to distribute these 6 pencils is: [ \binom{6 + 4 - 1}{4 - 1} = \binom{9}{3} ]
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Calculate the binomial coefficient: The binomial coefficient (\binom{9}{3}) can be calculated as: [ \binom{9}{3} = \frac{9}{3!(9-3)!} = \frac{9}{36} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 ]
Therefore, there are 84 ways to distribute 10 identical pencils to 4 students such that each student receives at least one pencil124.