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Prove that the number of ways in which p positive and n negative signs may be placed in a row so that no two negative signs are together is p+1 C n

Prove that the number of ways in which p positive and n negative signs may be placed in a row so that no two negative signs are together is p+1Cn

Grade:11

1 Answers

Arun
25750 Points
5 years ago
Number of ways in which p positive and n negative signs may be placed in a rowso that no two negative signs shall be together ?. More than an answer, I needthe reasoning behind the answer. Suppose you have p=10 positive signs and n=4 negative signs: + + + + + + + + + + 1 2 3 4 5 6 7 8 9 10 11Since negative signs cannot come together, we can only put one negativesign before any of the 10 positive signs, plus we can also place a negativesign after the 10th positive sign. So there are 11 places we can insert anegative sign, and we must choose 4 of those to place the 4 negative signs:So the answer for this specific case is 11C4The reasoning is the same for any number p of positive signs and any numbern of negative signs. We can only put one negative sign before any of the ppositive signs, plus we can also place a negative sign after the pth positivesign. So there are p+1 places we can insert a negative sign, and we mustchoose n of those to place the n negative signs:So the answer is "(p+1) choose n" which is also called "the number ofcombinations of p+1 things taken n at a time".It is calculated as (p+1)Cn

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