How many five-letter words beginning and ending with a vowel, without any letter repeated, can be formed from the word “personnel”?<\/p>\n\n\n\n
The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n To solve this problem, we will first identify the available letters in the word “personnel,” then calculate how many five-letter words can be formed that begin and end with a vowel, with no repeated letters.<\/p>\n\n\n\n The word “personnel” contains the following distinct letters: p, e, r, s, o, n, l. The vowels in the word are e, o, and e (with e repeating). Therefore, the unique vowels are e<\/strong> and o<\/strong>.<\/p>\n\n\n\n We have two distinct vowels (e and o) available for the first and last positions. The possible choices for the first and last letters are:<\/p>\n\n\n\n Thus, there are 2 possible ways<\/strong> to choose the first and last vowels.<\/p>\n\n\n\n Once the first and last letters are selected, we must pick 3 letters from the remaining available consonants and vowels. The remaining letters from the word “personnel” are: p, r, s, n, l. There are 5 distinct consonants, and these are the only letters available for the middle three positions.<\/p>\n\n\n\n For the middle three positions, we need to select and arrange 3 letters from the 5 remaining consonants. The number of ways to choose and arrange 3 letters out of 5 is given by the permutation formula: P(5,3)=5!(5\u22123)!=5\u00d74\u00d731=60P(5, 3) = \\frac{5!}{(5 – 3)!} = \\frac{5 \\times 4 \\times 3}{1} = 60<\/p>\n\n\n\n Now, we multiply the number of ways to choose the vowels for the first and last positions (2 ways) by the number of ways to arrange the remaining 3 letters (60 ways): Total number of words=2\u00d760=120\\text{Total number of words} = 2 \\times 60 = 120<\/p>\n\n\n\n Thus, 120<\/strong> distinct five-letter words can be formed from the word “personnel” that begin and end with a vowel, without any repeated letters.<\/p>\n","protected":false},"excerpt":{"rendered":" How many five-letter words beginning and ending with a vowel, without any letter repeated, can be formed from the word “personnel”? The correct answer and explanation is: To solve this problem, we will first identify the available letters in the word “personnel,” then calculate how many five-letter words can be formed that begin and end […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-45045","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45045","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=45045"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45045\/revisions"}],"predecessor-version":[{"id":45047,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45045\/revisions\/45047"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=45045"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=45045"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=45045"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Identifying Available Letters<\/h3>\n\n\n\n
Step 2: Conditions for the Word<\/h3>\n\n\n\n
\n
Step 3: Choosing the First and Last Letters<\/h3>\n\n\n\n
\n
Step 4: Choosing the Remaining Three Letters<\/h3>\n\n\n\n
Step 5: Total Number of Words<\/h3>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n