{"id":24684,"date":"2025-06-18T16:54:52","date_gmt":"2025-06-18T16:54:52","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=24684"},"modified":"2025-06-18T16:54:53","modified_gmt":"2025-06-18T16:54:53","slug":"in-how-many-ways-can-14-different-pizza-toppings-be-chosen-from-16-available-toppings","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/in-how-many-ways-can-14-different-pizza-toppings-be-chosen-from-16-available-toppings\/","title":{"rendered":"In how many ways can 14 different pizza toppings be chosen from 16 available toppings"},"content":{"rendered":"\n

In how many ways can 14 different pizza toppings be chosen from 16 available toppings?<\/p>\n\n\n\n

The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n

To find the number of ways to choose 14 different pizza toppings from 16 available toppings, we use combinations<\/strong> because the order of choosing toppings does not<\/strong> matter.<\/p>\n\n\n\n

\n\n\n\n

Step 1: Understand the problem type<\/h3>\n\n\n\n
    \n
  • You have 16 different toppings.<\/li>\n\n\n\n
  • You want to select 14 toppings.<\/li>\n\n\n\n
  • The order in which you select the toppings doesn’t matter (choosing topping A then B is the same as choosing B then A).<\/li>\n\n\n\n
  • This is a combination problem, not a permutation problem.<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    Step 2: Use the combination formula<\/h3>\n\n\n\n

    The formula for combinations is: (nr)=n!r!(n\u2212r)!\\binom{n}{r} = \\frac{n!}{r!(n-r)!}<\/p>\n\n\n\n

    where<\/p>\n\n\n\n

      \n
    • nn = total number of items (16 toppings),<\/li>\n\n\n\n
    • rr = number of items to choose (14 toppings),<\/li>\n\n\n\n
    • !! denotes factorial (the product of all positive integers up to that number).<\/li>\n<\/ul>\n\n\n\n
      \n\n\n\n

      Step 3: Calculate the number of ways<\/h3>\n\n\n\n

      (1614)=16!14!\u00d7(16\u221214)!=16!14!\u00d72!\\binom{16}{14} = \\frac{16!}{14! \\times (16 – 14)!} = \\frac{16!}{14! \\times 2!}<\/p>\n\n\n\n

      Since 2!=22! = 2, and 16!\/14!=16\u00d715\u00d714!\/14!=16\u00d71516! \/ 14! = 16 \\times 15 \\times 14! \/ 14! = 16 \\times 15, we get: (1614)=16\u00d7152=2402=120\\binom{16}{14} = \\frac{16 \\times 15}{2} = \\frac{240}{2} = 120<\/p>\n\n\n\n

      \n\n\n\n

      Final answer:<\/h3>\n\n\n\n

      120\\boxed{120}<\/p>\n\n\n\n

      There are 120<\/strong> different ways to choose 14 toppings out of 16.<\/p>\n\n\n\n

      \n\n\n\n

      Explanation in detail:<\/h3>\n\n\n\n

      When choosing toppings where order does not matter, combinations help us avoid counting duplicates caused by different orders of the same selection. For example, picking toppings A and B is the same as picking B and A.<\/p>\n\n\n\n

      Choosing 14 toppings from 16 is the same as excluding 2 toppings from 16 (because 16 – 14 = 2). Hence, the number of ways to choose 14 toppings is also equal to the number of ways to choose which 2 toppings to leave out. This symmetry is why (1614)=(162)\\binom{16}{14} = \\binom{16}{2}.<\/p>\n\n\n\n

      Factorials grow very fast, but using factorial properties simplifies calculations. For example, 16!\/14!=16\u00d71516! \/ 14! = 16 \\times 15 because all the terms from 14 down to 1 cancel out.<\/p>\n\n\n\n

      Thus, this approach efficiently gives the total number of ways to pick 14 toppings from 16, and the answer is 120 possible unique combinations.<\/p>\n","protected":false},"excerpt":{"rendered":"

      In how many ways can 14 different pizza toppings be chosen from 16 available toppings? The correct answer and explanation is: To find the number of ways to choose 14 different pizza toppings from 16 available toppings, we use combinations because the order of choosing toppings does not matter. Step 1: Understand the problem type […]<\/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-24684","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24684","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=24684"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24684\/revisions"}],"predecessor-version":[{"id":24685,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/24684\/revisions\/24685"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=24684"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=24684"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=24684"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}