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Seven-letter words can be formed without repetition from the letters of the word INCLUDE; 5040. How many have the letters N and D separated by exactly two letters?

Feb 15, 2020

$$\text{Select the slot that N and D will occupy}\\ \text{There are 4 of these}\\ \text{Select the ordering of N and D within the slot}\\ \text{There are two arrangements}\\ \text{There are now 5 unique remaining letters}\\ \text{These can be arranged 5! ways in the remaining slots}\\ \text{So there are 4 \cdot 2 \cdot 5! = 960 valid arrangements}\\ \text{There are a total of 7!=5040 possible arrangements}\\ p = \dfrac{960}{5040} = \dfrac{4}{21}$$