Evaluate three 𝑃 three plus 11𝑃 four plus nine 𝑃 five.
We begin by recalling that the notation 𝑃 in this question denote permutations, where 𝑛𝑃𝑘 represents the number of different ways to order 𝑘 objects from 𝑛 total distinct objects. This value of 𝑛𝑃𝑘 is equal to 𝑛 factorial divided by 𝑛 minus 𝑘 factorial. This means that three 𝑃 three is equal to three factorial divided by zero factorial. Recalling that zero factorial equals one, this is simply equal to three factorial. And since this is equal to three multiplied by two multiplied by one, three 𝑃 three is equal to six.
Next, we have 11𝑃 four, which is equal to 11 factorial divided by seven factorial. 11 factorial can be rewritten as 11 multiplied by 10 multiplied by nine multiplied by eight multiplied by seven factorial. Dividing the numerator and denominator by seven factorial, 11𝑃 four is equal to 11 multiplied by 10 multiplied by nine multiplied by eight. This is equal to 7,920.
Finally, we have nine 𝑃 five, which is equal to nine factorial divided by four factorial. Using the same method as with our second term, this is equal to nine multiplied by eight multiplied by seven multiplied by six multiplied by five, which is equal to 15,120. We need to find the sum of our three values. Six plus 7,920 plus 15,120 is equal to 23,046. This is the value of three 𝑃 three plus 11𝑃 four plus nine 𝑃 five.