Find the number of distinct arrangements of the 11 letters in reinterpret
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The word ARRANGEMENT has $11$ letters, not all of them distinct. Imagine that they are written on little Scrabble squares. And suppose we have $11$ consecutive slots into which to put these squares. Show
There are $\dbinom{11}{2}$ ways to choose the slots where the two A's will go. For each of these ways, there are $\dbinom{9}{2}$ ways to decide where the two R's will go. For every decision about the A's and R's, there are $\dbinom{7}{2}$ ways to decide where the N's will go. Similarly, there are now $\dbinom{5}{2}$ ways to decide where the E's will go. That leaves $3$ gaps, and $3$ singleton letters, which can be arranged in $3!$ ways, for a total of $$\binom{11}{2}\binom{9}{2}\binom{7}{2}\binom{5}{2}3!.$$ If they were all distinguishable then the number of distinct arrangements would be#10!#. We can make them distinguishable by adding subscripts:
If we remove the subscripts from the letter#O#'s, then it no longer makes any difference what order the#O#'s are in and we find that#1/(2!) = 1/2#of our#10!#arrangements are identical to the other half. So there are#(10!)/(2!)#possible arrangements of the letters:
If we remove the subscripts from the letter#K#'s a similar thing happens and we are left with half again. So there are#(10!)/(2!2!)#possible arrangements of the letters:
Finally, since#E_1#,#E_2#and#E_3#can be arranged in#3!#possible orders, then when we remove the subscripts from the#E#'s there are#(10!)/(2!2!3!)#distinct arrangements of the letters: Microsoft and Sony are both battling behind the scenes over the Activision Blizzard deal, and Microsoft is no longer pulling its punches with regulatorsBy Tom Warren / @tomwarren Oct 12, 2022, 6:23 PM UTC| Share this story
Illustration by Alex Castro / The Verge Microsoft isn’t happy with Sony and the UK’s Competition and Markets Authority. The UK regulator signaled an in-depth review of Microsoft’s $68.7 billion deal to acquire Activision Blizzard last month, and the CMA has now published its full 76-page report (PDF) on its findings. The CMA says it has concerns that Microsoft’s Activision Blizzard deal could lessen competition in game consoles, subscriptions, and cloud gaming, but Microsoft thinks the regulator has simply been listening to Sony’s lawyers too much. Microsoft pleaded for its deal on the day of the Phase 2 decision last month, but now the gloves are well and truly off. Microsoft describes the CMA’s concerns as “misplaced” and says that the regulator “adopts Sony’s complaints without considering the potential harm to consumers” and “incorrectly relies on self-serving statements by Sony which significantly exaggerate the importance of Call of Duty.” Microsoft even accuses the CMA of adopting “Sony’s complaints without the appropriate level of critical review,” suggesting that the regulator is simply just listening too much to what Sony has to say. At the heart of all the back and forth is access to Call of Duty and concerns around the future of game subscriptions. “The CMA recognizes that ABK’s newest games are not currently available on any subscription service on the day of release but considers that this may change as subscription services continue to grow,” says the UK regulator. “After the Merger, Microsoft would gain control of this important input and could use it to harm the competitiveness of its rivals.” Microsoft’s full response to the CMA, seen by The Verge, also includes parts where the company tries to, comically, make it look like it somehow sucks at gaming and it can’t compete. Microsoft says Xbox “is in last place in console” and “seventh place in PC” and “nowhere in mobile game distribution globally,” and Microsoft argues it has no reason to harm or degrade rival cloud gaming services as it wants to “encourage the major shift in consumer behavior required for cloud gaming to succeed.” Microsoft might well be in last place in console sales during the previous generation, but it’s certainly investing billions of dollars to ensure any future Xbox sales aren’t less than half of the PlayStation and that its Xbox Game Pass bet pays off. Sony and Microsoft have also been battling it out over Call of Duty, and the CMA recognizes this by revealing it’s concerned about Sony’s future revenues related to Call of Duty. “PlayStation currently has a larger share of the console gaming market than Xbox, but the CMA considers that Call of Duty is sufficiently important that losing access to it (or losing access on competitive terms) could significantly impact Sony’s revenues and user base.”
Call of Duty is at the center of Sony and Microsoft’s battles.Image: Activision Sony has shown how significant Call of Duty is after it labeled Microsoft’s offer to keep Call of Duty on PlayStation “inadequate on many levels.” The Verge revealed last month that Microsoft Gaming CEO and Xbox chief Phil Spencer made a written commitment to PlayStation head Jim Ryan earlier this year to keep Call of Duty on PlayStation for “several more years” beyond the existing marketing deal Sony has with Activision. “After almost 20 years of Call of Duty on PlayStation, their proposal was inadequate on many levels and failed to take account of the impact on our gamers,” said PlayStation head Jim Ryan in response. Now Microsoft says keeping Call of Duty on PlayStation is a “commercial imperative for the Xbox business and the economics of the transaction.” Microsoft says it would put revenue at risk if it pulled Call of Duty from PlayStation and that “Microsoft has been clear that it is counting on revenues from the distribution of Activision Blizzard games on Sony PlayStation.” Microsoft also accuses Sony of not welcoming competition from Xbox Game Pass and that Sony has decided to block Game Pass on PlayStation. “This increased competition has not been welcomed by the market leader Sony, which has elected to protect its revenues from sales of newly released games, rather than offer gamers the choice of accessing them via its subscription, PlayStation Plus.” This comes just months after Microsoft claimed, in legal filings, that Sony pays for “blocking rights” to keep games off Xbox Game Pass. If the UK battles are anything to go by, this acquisition could get messy as Microsoft and Sony battle it out behind the scenes to sway regulators. Microsoft even has a dedicated website to highlight its arguments as it seeks to convince regulators that its giant deal isn’t a bad one for gamers. We’re still months away from final regulator decisions, but get ready for this battle to continue to spill out onto the internet’s streets. Join the conversation Most Popular
Verge Deals/ Sign up for Verge Deals to get deals on products we've tested sent to your inbox daily. Email (required)Sign up By submitting your email, you agree to our Terms and Privacy Notice. This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply. How many different arrangements are there for the 11 letters in the word combination?Answer: COMBINATION can be arranged in 4,989,600 different ways if it is eleven letters and only use each letter once.
What is the total number of different arrangements of the 11 letters in the word caterpillar?The answer is 6 * 8! or 241,920. Total no. of different arrangements of the 11 letters in the word caterpillar = 11!
How many distinct arrangements are there in the word reverse?different arrangements, and letters V and S can be arranged in 2 different ways (VS or SV). So there are 60⋅2=120 arrangements where V is together with S.
How many arrangements are there of the letters in bookkeeper?Answer and Explanation: There are 151200 distinct permutations that can be formed using the letters of the word 'BOOKKEEPER'. For the word 'BOOKKEEPER' there are 10 letters.
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