In how many ways can 5 identical black balls and 7 identical white balls be arranged in a row so that no 2 black balls are together
Choose the correct choice from the following choices
A) 56  B) 64  C) 65  D) 316  E) 560
Consider the possible arrangements:   WBWBWBWBWBWW or BWBWBWBWBWWW or WWBWBWBWBWBW or WWWBWBWBWBWB   Clearly, we are concerned about not placing 2 B's together, whites can be placed however we want.   So, it is like _W_W_W_W_W_W_W_ where black ball can be placed anywhere on the blank spaces, and this way, there never will be a case where 2 black balls fall together.   56s there are 8 black spaces where 5 black balls can be placed, thus, the combination turns out as 8C5 (again, we are not concerned about the placement of white balls as we have already "fixed" their positions in the diagram above)   56lso, we achieved at 8C5 as arranging 5 black balls on 8 different spots =&gt; 8!/5!3! (Total combination of available spaces = 8!; choosing 5 spaces out of them = 5!, arranging 3 leftover spaces = 3!) = 56   Hence answer is 56
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In how many ways can 5 identical black balls and 7 identical white balls be arranged in a row so that no 2 black balls are together