Please have a look at question 1 first. Can any one please tell me how to understand the method used in the second question?
Many thanks.
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1Choose $3$ out of $10$ places for the letter A, choose $4$ out of the remaining $7$ places for the letter B, choose $3$ out of the remaining $3$ places for the letter C. – barak manos Mar 13 '15 at 07:07
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possible duplicate of How many different words can be formed using all the letters of the word GOOGOLPLEX? – Marc van Leeuwen Mar 13 '15 at 12:06
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thanks barak, you have been most helpful. that is exactly what I need to know. – Yilin Wang Mar 14 '15 at 06:51
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There are 10 positions and you can think of this question in 3 stages and multiply the number of choices at each stage to get the final answer.
Stage 1: choose where to place the 3 letters A: $\binom{10}{3}$ choices (10 places, 3 indistinguishable objects). Stage 2: choose where to place the 4 letters B: $\binom{7}{4}$ choices (7 places left, 4 indistinguishable objects) Stage 3: choose where to place the 3 letters C: $\binom{3}{3}$ choices (3 places left, 3 indistinguishable objects, no choice actually).
You can observe that your answer is independent of the order of choices you make about the stages, by writing out the binomial coefficients and seeing the cancellations. So $\binom{10}{4}\binom{7}{3}\binom{3}{3}$ as well as $\binom{10}{3}\binom{7}{3}\binom{4}{4}$ give the same numerical answer.
As an exercise, you can write a general expression for $n$ total slots for $v$ distinct letters, and $k_i$ instances of each letter, where $k_1+k_2+...+k_v=n$ and simplify it.
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